TT HE XIV C
ONFERENCE ON
FAM
AND E VENTOLOGY OF M ULTIVARIATE S TATISTICS , K
RASNOYARSK , S
IBERIA , R
USSIA , 2015
Elements of the Kopula (eventological copula) theory
Oleg Yu. Vorobyev
Institute of Mathematics and Computer ScienceSiberian Federal UniversityKrasnoyarsk
Abstract.
New in the probability theory and even-tology theory, the concept of Kopula (eventologicalcopula) is introduced . The theorem on the charac-terization of the sets of events by Kopula is proved,which serves as the eventological pre-image of thewell-known Sclarβs theorem on copulas (1959). TheKopulas of doublets and triplets of events are given,as well as of some π -sets of events. Keywords.
Eventology, probability, Kolmogorovevent, event, set of events, Kopula (eventological cop-ula), Kopula characterizing a set of events.
Long time ago and little by little, the incentivefor this work materialized in the theory of sets ofevents, eventology [1], where the need to locate theclasses of event-probability distributions (e.p.d.) ofthe sets of events (s.e.), which were so arbitraryand spacious to be able without let or hindrance todeal with the relationships between pairs, triples,quadruplets, etc., of events, in other words, to un-derstand the structure of statistical dependenciesand generalities between events from some s.e. Asimilar need is perhaps the only one that has al-ways fueled the development of the probability the-ory and statistics, which in one way or another aretheories of studying and evaluating the structuresof statistical dependencies and generalities in thedistributions of sets of events.The classical copula theory [2, 3, 4, 5], existing sincethe 50s of the last century, allows us to constructclasses of joint distribution functions that havegiven marginal distribution functions. In eventol-ogy, the theory of sets of events, proposed in thepaper the theory of Kopula (eventological copula) allows us to solve a similar problem β to buildclasses of e.p.dβs of sets of events whose events hap-pen with given probabilities of marginal events . c β Oleg Vorobyev (ed.), Proc. XIV FAMEMSβ2015, Krasnoyarsk: SFU To distinguish the quite differently deο¬ned notion of an eventological copula from the classical concept of copula in thesense of Sklar (1959), the following radical terminology withcapital βKβ is used:
Kopula = eventological copula; π -Kopula =eventological π -copula. Editing the text of November 12, 2017.
We formulate the general statement of the problemof the π -Kopula theory for π -sets of events. If inthe classical theory the copula is the tool for select-ing some family of joint d.f.βs of a set of random vari-ables from the set of all d.f.βs with given marginald.f.βs, then in the eventological theory the Kopula isthe tool for selecting a family of e.p.d.βs of the 1stkind of the set events from the set of all e.p.d.βs withthe given probabilities of marginal events.However, unlike the classical d.f.βs the functions ofe.p.d.βs of the 1st kind of the π -s.e. X are functionsthat are deο¬ned as the sets { π ( π// X ) , π β X } (1.1)of all its π values, probabilities of the 1st kind π ( π// X ) , on the set of all subsets of this π -s.e.So, letβs clarify, Kopula is the tool for selecting afamily of sets of the form (1.1) from the set of allsets with given probabilities of marginal events.To specify a family of sets of π probabilities ofthe 1st kind (1.1), it is necessary and suο¬cient tospecify a family of sets from the π β parameters,since all the probabilities in each set must be non-negative and give in the sum of one. And to spec-ify a family of sets of π probability values of the1st kind (1.1) with given probabilities of marginalevents that form the X -set Λ π = { π π₯ , π₯ β X } , (1.2)it is necessary and suο¬cient to specify a family ofsets from the π β π β parameters, since for eachcollection there must be another π constraints forevents π₯ β X : βοΈ π₯ β π β X π ( π// X ) = π π₯ . (1.3)Therefore, βto deο¬ne the family of functions ofe.p.d.βs of the 1st kind with given probabilities ofmarginal eventsβ means βto deο¬ne a family of setsfrom the π β π β parametersβ as sets of functionsof these probabilities. Eventological theory shouldsolve this problem with the help of a convenienttool, the Kopula, which allows us to deο¬ne a fam-ily of sets from the π β π β parameters as sets a r X i v : . [ s t a t . O T ] F e b OROBYEV of functions from marginal probabilities, which inturn can be made dependent on a number of auxil-iary parameters.In general, the π -Kopula in the eventological the-ory is an instrument for deο¬ning the family ofprobability distributions of the 1st kind of the π -s.e., in the form of a family of π -set of functions ofthe probabilities of their π marginal events. The main results of this paper are presented ina rather rigorous mathematical manner. And al-though the deο¬nitions, statements and proofs areprovided with examples and illustrations, in orderto visualize the ideas underlying the Kopula theory,in my opinion, a number of preliminary explana-tions in a less strict context, which are collected inseveral prolegomena, may be necessary.If in some set of events X , some events from thesubset π β X are replaced by their complements,then we get a new set of events X ( π | π ) = π + ( X β π ) ( π ) , which is called the π -phenomenon of s.e. X .The set of all such π -phenomena for π β X iscalled the X -phenomenon-dom of s.e. X . In [6]a rather distinct theory of set-phenomena and thephenomenon-dom of some s.e.Similarly, the theory of set-phenomena [6] de-ο¬nes the phenomena and phenomenon-dom ofthe set Λ π = { π π₯ , π₯ β X } of the probabilities ofmarginal events π₯ β X : the π -phenomenon Λ π ( π | π ) = { π π₯ , π₯ β π } + { β π π₯ , π₯ β X β π } of the set ofmarginal probabilities Λ π is obtained by replacingthe marginal probabilities π π₯ by complementarymarginal probabilities β π π₯ when π₯ β X β π . Prolegomenon 1 (set-phenomenon of a set of eventsand a set of probabilities of events).
The main con-clusion of the above theory is obvious: probabilitydistribution of the s.e. X characterizes the probabil-ity distribution and the set of marginal probabilitiesof each set-phenomenon from its X -phenomenon-dom . Prolegomenon 2 (set-phenomenal transformation). X ( π | π ) , X ( π | π ) set-phenomena of the setof events X their probability distributions are relatedto each other set-phenomenal transformations . Λ π ( π | π ) , Λ π ( π | π ) set-phenomena of the set ofmarginal probabilities Λ π events from X are also in-terconnected by set-phenomenal transformations .The event π₯ β X is called half-rare [6] if the proba-bility π π₯ = P ( π₯ ) with which it happens is not morethan half: π π₯ (cid:54) / . If all events from the s.e. X arehalf-rare, we speak of a set of half-rare events, or ahalf-rare s.e. Prolegomenon 3 (sets of half-rare events and itsKopulas).
1) It is not diο¬cult to guess that for any s.e. X , in the X phenomenon-dom of the sets of events X ( π | π ) , π β X , and in the X -phenomenon-dom of the sets of its marginal probabilities Λ π ( π | π ) , π β X , thereis always a half-rare set-phenomenon. If, in addition,there are no events in X happening with probability / , then such a half-rare set-phenomenon is unique.2) A Kopula of some family of half-rare π -s.e.βs isgenerated by π functions from half-rare variablesdeο¬ned on the half-hypercube [0 , / π with valuesfrom [0 , that are continued by the set-phenomenaltransformations of the half-rare variables to the cor-responding half-hypercubes, all together completelyο¬lling the unit hypercube . Prolegomenon 4 (an invariance of the copula withrespect to the order of half-rare events).
Our task isto construct a Kopula of a family of arbitrary (un-ordered) sets of half-rare events, i.e. a 1-function,the arguments of which form an unordered set ofprobabilities of their marginal events. Therefore, itis natural to require such a function to be invariantwith respect to the order of its arguments; withrespect to the order of events in these sets. In otherwords, it is natural to consider this 1-function as afunction of a set of arguments, rather than a vectorof arguments with ordered components, as it isusually assumed.
Prolegomenon 5 (insertable sets of half-rare eventsand a frame half-rare event).
Two insertable sets ofhalf-rare events for a given set of half-rare events X = { π₯ } + π³ with the frame half-rare event π₯ β X ,happening with the highest probability among allevents from X , are two sets of half-rare events π³ β² = { π₯ } ( β© ) π³ and π³ β²β² = { π₯ π } ( β© ) π³ that partitionthe set of other events π³ = X β { π₯ } into two: π³ = π³ β² (+) π³ β²β² . The events of one of them, π³ β² ,are contained in the frame half-rare event π₯ , andthe events of the other, π³ β²β² , are contained in itscomplement π₯ π = β¦ β π₯ . Prolegomenon 6 (the insertable sets of events andconditional e.p.d.βs of a set of events with respectto the frame event and its complement).
Conditionale.p.d.βs of the 1st kind of one s.e. X with respectto the other s.e. Y are deο¬ned in the traditionalway [1]. However, until now attempts to deο¬nesuch a βconditionalβ s.e., which would have givena conditional e.p.d. of the 1st kind, turned out tobe completely impractical [7]. The concept of twoinsertable s.e.βs in a frame event is a well-deο¬nedβersatzβ of such βconditionalβ s.e.βs. The e.p.d.βsof this βersatzβ, although they do not coincidewith two conditional e.p.d.βs of the 1st kind withrespect to the frame event and its complement,but they are fully characterized by them. Theconverse is also true: the e.p.d.βs of two frame s.e.βscharacterize the corresponding two conditionede.p.d.βs of the 1st kind. Prolegomenon 7 (a frame method of constructinga Kopula of an arbitrary set of half-rare events). A frame method constructs a Kopula of a family ofarbitrary sets of half-rare events on the basis ofa conditional scheme by means of a recurrenceformula via conditional e.p.d.βs concerning theframe event and its complement. A recurrent HE XIV FAMEMSβ2015 C
ONFERENCE formula associates this Kopula with two Kopulas offamilies of their insertable sets of half-rare eventsof smaller dimension, which are characterized bythe corresponding conditional e.p.d.βs of the 1stkind (see Prolegomenon 6).
Prolegomenon 8 (a set-phenomenal transformation ofa half-rare Copula to an arbitrary one).
To constructthe Kopula of a family of arbitrary s.e.βs it is enoughto construct the Kopula of the family of their half-rare set-phenomena and apply a set-phenomenaltransformation to this Kopula.
Prolegomenon 9 (Cartesian representation of the π -Kopula in R π ). It follows from the Prolegomenon4 that the Cartesian representation of the π -Kopulain R π should be a symmetric function of π orderedvariables, marginal probabilities of events from π -s.e. X , which is deο¬ned on the π -dimensional unithypercube [0 , π . The Cartesian representation ofthe π -Kopula is based on the fact that its symmetricimage takes the same values on all permutations ofits arguments, that is, is deο¬ned by the permutationof π events group. Moreover, the value of sucha symmetric function on an arbitrary π -vector Β― π€ = { π€ , ..., π€ π } β [0 , π is equal to the valueof the π -Kopula on an ordered X -set of marginalprobabilities of half-rare events Λ π = { π π₯ , π₯ β X } ,the ordered half-rare projection of the π -vector Β― π€ ,the order of the variables in which is deο¬ned by an π -permutation π Β― π€ that has the components of thehalf-rare projection Β― π€ * in decreasing order, where π€ * π = {οΈ π€ π , π€ π (cid:54) / , β π€ π , π€ π > / (1.4) are components of the π -vector Β― π€ * of the half-rareprojection of the π -vector Β― π€ , π = 1 , ..., π . As a result,the ordered half-rare X is the set of marginal prob-abilities Λ π = Λ π ( Β― π€ ) , on which the π -Kopula takes thesame value as a symmetric function on Β― π€ , is given bythe formula Λ π ( Β― π€ ) = π Β― π€ ( Β― π€ * ) , (1.5) which deο¬nes the Cartesian representation of the π -Kopula in R π for each Β― π€ β [0 , π . We consider the general probability space of Kol-mogorov events (β¦ , π (cid:102) , P ) , some particular proba-bility space of events (β¦ , π , P ) and the π - set of events( π -s.e.) X β π with the event-probability distribu-tion (e.p.d. ) of the 1st kind π ( X ) = { π ( π// X ) : π β X } , The abbreviations: e.p.d. and e.c.d. are used for the event-probability distribution and for the event-covariance distribu-tion . and of the second kind π X = { π π// X : π β X } , which, recall, are related to each other by the Mo-bius inversion formulas: π π// X = βοΈ π β π π ( π // X ) ,π ( π// X ) = βοΈ π β π ( β | π |β| π | π π// X . Definition 1 (set-phenomena of a s.e. and itsphenomenon-dom).
Every π -s.e. X β π generates itsown X -phenomenon-dom , deο¬ned as a π -family ( π | X ) = {οΈ X ( π | π ) , π β X }οΈ , (2.1)composed of π -s.e. in the form X ( π | π// X ) = X ( π | π ) = π + ( X β π ) ( π ) β π , which for each π β X is called its set-phenomen [6], more precisely , π -phenomen , where π ( π ) = { π₯ π : π₯ β π } is an -complement of the s.e. π β X .We also recall that probabilities of the second kind π π₯ = π { π₯ } = P ββ βοΈ π₯ β{ π₯ }β X π₯ ββ = P ( π₯ ) are probabilities of marginal events from { π₯ } β X ( marginal probabilities ), probabilities of the secondkind π π₯π¦ = π { π₯,π¦ } = P ββ βοΈ π§ β{ π₯,π¦ }β X π§ ββ = P ( π₯ β© π¦ ) are probabilities of double intersections of eventsfrom { π₯, π¦ } β X , and probabilities of the secondkind π π π = P ββ βοΈ π₯ β π π β X π₯ ββ are probabilities of π -intersections of events from π π β X , where | π π | = π ; Definition 2 (set-phenomena of the set of probabil-ities of events from a s.e. and its phenomen-dom).
The π -set of probabilities of events from X Λ π = { π π₯ : π₯ β X } also generates its X -phenomen-dom , the π -totality ( π | Λ π ) = {οΈ Λ π ( π | π// X ) , π β X }οΈ , (2.2) OROBYEV composed of π -sets in the form Λ π ( π | π// X ) = {οΈ π π§ : π§ β X ( π | π ) }οΈ , and deο¬ned for π β X as the π -set of probabilitiesof events from π -phenomenon X ( π | π ) of the s.e. X where for π π§ β Λ π ( π | π// X ) π π§ = {οΈ π π₯ , π§ = π₯ β π, β π π₯ π§ = π₯ π β π ( π ) . In particular, for π = X Λ π ( π | X // X ) = { π π₯ : π₯ β X } = Λ π. We denote by π : β¨οΈ π₯ β X [0 , π₯ β R +0 (2.3)a nonnegative bounded numerical function de-ο¬ned on the set-product [8], X -hypercube [0 , β X = β¨οΈ π₯ β X [0 , π₯ . Arguments of π form the π -set Λ π€ = { π€ π₯ : π₯ β X } β [0 , β X which generates its own X -phenomenon-dom , the π -totality ( π | Λ π€ ) = {οΈ Λ π€ ( π | π// X ) , π β X }οΈ (2.4)of π -sets of arguments: Λ π€ ( π | π// X ) = {οΈ π€ π§ : π§ β X ( π | π ) }οΈ (2.5)where for π€ π§ β Λ π€ ( π | π// X ) π€ π§ = {οΈ π€ π₯ , π§ = π₯ β π, β π€ π₯ π§ = π₯ π β π ( π ) . Let Ξ¨ X = {οΈ π ββ π : [0 , β X β R +0 }οΈ (2.6)be the family of all the nonnegative bounded nu-merical functions on the X -hypercube. Definition 3 ( normalized function on the X -hypercube ). A function π β Ξ¨ X is called normalized on the X -hypercube if for each Λ π€ β [0 , β X βοΈ π β X π (οΈ Λ π€ ( π | π// X ) )οΈ = 1 , (2.7)i.e., the sum of its values on all the π -sets of argu-ments from X -phenomenon-dom ( π | Λ π€ ) is one. Definition 4 ( a 1-function on the X -hypercube ). A function π β Ξ¨ X is called a on the X -hypercube if for all Λ π€ β [0 , β X π₯ - marginalequalities are satisο¬ed for all π₯ β X : βοΈ π₯ β π β X π (οΈ Λ π€ ( π | π// X ) )οΈ = π€ π₯ , (2.8)i.e., the sum of its values on π₯ -halves of π -sets ofarguments from the X -phenomenon-dom ( π | Λ π€ ) is π€ π₯ .Denote by Ξ¨ X = β§β¨β© π β Ξ¨ X : βοΈ π β X π (οΈ Λ π€ ( π | π// X ) )οΈ = 1; Λ π€ β [0 , β X β«β¬β the family of functions, normalized on the X -hypercube; and by Ξ¨ X = β§β¨β© π β Ξ¨ X : βοΈ π₯ β π β X π (οΈ Λ π€ ( π | π// X ) )οΈ = π€ π₯ ; Λ π€ β [0 , β X β«β¬β the family of 1-functions on the X -hypercube. properties of 1-functions on the { π₯, π¦ } -square ). A strict inclusion is fair: Ξ¨ { π₯,π¦ } β Ξ¨ { π₯,π¦ } . Proof . In other words, the lemma states: 1) if π β Ξ¨ { π₯,π¦ } is a 1-function on the { π₯, π¦ } -square then π β Ξ¨ { π₯,π¦ } is a normalized function on the { π₯, π¦ } -square; 2) among the normalized functions from Ξ¨ { π₯,π¦ } there is one which is not a 1-function. Butthis is obvious, as it is conο¬rmed by the followingsimple examples.First, indeed, for the doublet of events X = { π₯, π¦ } bythe deο¬nition of a 1-function, we have π ( π€ π₯ , π€ π¦ ) + π ( π€ π₯ , β π€ π¦ ) = π€ π₯ , (2.9) π ( π€ π₯ , π€ π¦ ) + π (1 β π€ π₯ , π€ π¦ ) = π€ π¦ , (2.10) π (1 β π€ π₯ , β π€ π¦ ) + π ( π€ π₯ , β π€ π¦ ) = 1 β π€ π¦ , (2.11) π (1 β π€ π₯ , β π€ π¦ ) + π (1 β π€ π₯ , π€ π¦ ) = 1 β π€ π₯ . (2.12)The sums (2.9) and (2.12) as well as the sums (2.10)and (2.11) as a result give π ( π€ π₯ , π€ π¦ ) + π ( π€ π₯ , β π€ π¦ )++ π (1 β π€ π₯ , β π€ π¦ ) + π ( π€ π₯ , β π€ π¦ ) = 1 , (2.13)i.e., π β Ξ¨ { π₯,π¦ } is a normalized function on the { π₯, π¦ } -square.Second, the function (see its graph in Fig. 1 ) π ( π€ π₯ , π€ π¦ ) = ( π€ π₯ + π€ π¦ ) / In this ο¬gure and others, which illustrate the doublets ofevents, the map of this function on a unit square is shown un-der the graph in conditional colors where the white color corre-sponds to the level 1/4. HE XIV FAMEMSβ2015 C
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Figure 1: The graph of the Cartesian representation of the normalizedfunction π ( π€ π₯ , π€ π¦ ) = ( π€ π₯ + π€ π¦ ) / on the { π₯, π¦ } -square from Ξ¨ { π₯,π¦ } which is not a 1-function. is normalized on the { π₯, π¦ } -square, since π ( π€ π₯ , π€ π¦ ) + π ( π€ π₯ , β π€ π¦ )+ π (1 β π€ π₯ , π€ π¦ ) + π (1 β π€ π₯ , β π€ π¦ ) == ( π€ π₯ + π€ π¦ ) / π€ π₯ + 1 β π€ π¦ ) / β π€ π₯ + π€ π¦ ) / β π€ π₯ + 1 β π€ π¦ ) / . However, it is not a 1-function, since π ( π€ π₯ , π€ π¦ ) + π ( π€ π₯ , β π€ π¦ ) = ( π€ π₯ + π€ π¦ ) / π€ π₯ + 1 β π€ π¦ ) / π€ π₯ / / ΜΈ = π€ π₯ ,π ( π€ π₯ , π€ π¦ ) + π (1 β π€ π₯ , π€ π¦ ) = ( π€ π₯ + π€ π¦ ) / β π€ π₯ + π€ π¦ ) / π€ π¦ / / ΜΈ = π€ π¦ . The lemma is proved.Of course, in the general case, for an arbitrary s.e. X the same lemma is fulο¬lled. Lemma 2 ( properties of 1-functions on the X -hypercube ). A strict inclusion is fair: Ξ¨ X β Ξ¨ X . Proof is similar.
Note 1 (a representation of a 1-function on the X -hypercube in the form of | X | -set of functions). Any 1-function π β Ξ¨ X on the X -hypercube [0 , β X for each Λ π€ β [0 , β X is represented in the form of | X | -set of the following functions: π ( Λ π€ ) = {οΈ π π ( Λ π€ ) , π β X }οΈ == {οΈ π (οΈ Λ π€ ( π | π// X ) )οΈ , π β X }οΈ . (2.14) Definition 5 (
Kopula ). The 1-functions K β Ξ¨ X β Ξ¨ X is called | X | -Kopulas of the s.e. X .As well as every 1-function (2.14), any | X | -Kopula ofthe s.e. X can be represented for Λ π€ β [0 , β X in theform of | X | -set of the following functions: K ( Λ π€ ) = {οΈ K π ( Λ π€ ) , π β X }οΈ == {οΈ K (οΈ Λ π€ ( π | π// X ) )οΈ , π β X }οΈ . (2.15) Note 2 (characteristic properties of Kopula).
Each Kopula K has two characteristic properties1) Kopula is nonnegative : K (οΈ Λ π€ ( π | π// X ) )οΈ (cid:62) (2.16)for π β X , since by deο¬nition K β Ξ¨ X ;2) Kopula ia satisο¬ed π₯ - marginal equalities : βοΈ π₯ β π β X K (οΈ Λ π€ ( π | π// X ) )οΈ = π€ π₯ (2.17)for π₯ β X , since by deο¬nition K β Ξ¨ X ;From (2.17) by Lemma 2 a probabilistic normaliza-tion of the Kopula follows: βοΈ π β X K (οΈ Λ π€ ( π | π// X ) )οΈ = 1 . (2.18)From (2.16) and (2.18) terrace-by-terrace probabilis-tic normalization of the Kopula follows: (cid:54) K (οΈ Λ π€ ( π | π// X ) )οΈ (cid:54) (2.19)for π β X . The eventological analogue and the preimage of thewell-known Sklar theorem on copulas [2] is the fol-lowing theorem.
Theorem 1 ( characterization of a s.e. by Kop-ula ). Let π = { π ( π// X ) : π β X } be the e.p.d. ofthe 1st kind of the s.e. X with X -set of probabilitiesof marginal events Λ π = { π π₯ : π₯ β X } β [0 , β X . Thenthere is a | X | -Kopula K β Ξ¨ X that deο¬nes a familyof e.p.d.βs of the 1st ο¬nd of the s.e. X . This familycontains the e.p.d. π , when Kopulaβs arguments see the footnote 1 on page 78. OROBYEV coincide with Λ π . In other words, the such Kopulathat foe all π β X π ( π// X ) = K π (Λ π ) = K (οΈ Λ π ( π | π// X ) )οΈ . (2.20) (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) Conversely, for any X -set of probabilities of marginalevents Λ π β [0 , β X and any | X | -Kopula K β Ξ¨ X , thefunction π = { π ( π// X ) : π β X } , deο¬ned by formulas(2.20) for π β X , is an e.p.d. of the 1st kind, whichcharacterizes the s.e. X with given X -set of the prob-abilities of marginal events Λ π . Proof is a direct consequence of the properties ofe.p.d. of the 1st kind of the s.e. X and the | X | -Kopula. First, if the e.p.d. of the 1st kind π = { π ( π// X ) : π β X } of some s.e. X with the X -set ofmarginal probabilities Λ π = { π π₯ : π₯ β X } is deο¬ned,then from properties of probabilities of the 1st kindit follows that for π₯ β X π π₯ = βοΈ π₯ β π β X π ( π// X ) , (2.21)i.e., the function K , deο¬ned by the e.p.d. of the1st kind π and formulas (2.3), satisο¬es π₯ -marginalequalities for π₯ β X : π π₯ = βοΈ π₯ β π β X K (Λ π ( π | π// X ) ) (2.22)(required for being a 1-function:: K β Ξ¨ X ) andserves as the | X | -Kopula.Second, if the function K is the | X | -Kopula, then byLemma 1: K β Ξ¨ X β Ξ¨ X , i.e., it is normalized and,consequently, by (2.20) the function π is normalizedtoo: βοΈ π β X π ( π// X ) . (2.23)In addition, from (2.20) and from the fact that the | X | -Kopula is a 1-function, (2.21) follows for all π₯ β X . Therefore, the function π is a e.p.d. of the 1stkind of the s.e. X with the X -set of marginal proba-bilities Λ π . The theorem is proved. Lemma 3 ( convex combination of Kopulas ). A con-vex combination of an arbitrary set of Kopulas of oneand the same s.e. is its Kopula too. Proof without tricks. Let X be a s.e., and K , . . . , K π (2.24)be some set of its Kopulas. Let us prove that theirconvex combination K = π βοΈ π =1 πΌ π K π (2.25) (where, of course, πΌ + . . . + πΌ π = 1 , πΌ π (cid:62) , π =1 , . . . , π ) is also a Kopula. For this it suο¬ces to provethat K is a 1-function. In other words, that for π₯ β X βοΈ π₯ β π β X K (Λ π ( π | π// X ) ) = π π₯ . (2.26)Since each Kopula from the set (2.24) has a prop-erty of a 1-function, then for π₯ β X we get what isrequired: βοΈ π₯ β π β X K (Λ π ( π | π// X ) ) == π βοΈ π =1 πΌ π βοΈ π₯ β π β X K π (Λ π ( π | π// X ) ) == π βοΈ π =1 πΌ π π π₯ = π π₯ . (2.27) Corollary 1 ( convex combination of Kopulas ). For every set of events X the space of 1-functions Ξ¨ X , as well as the space of its Kopulas, is a convexmanifold. Without set-phenomenon transformations andvariable transformations, analytic work on setsof half-rare events (s.h-r.e.βs) (see [6]) and sets oftheir marginal probabilities, is unlikely to be ef-fective. However, in speciο¬c calculations at ο¬rst,because of their unaccustomedness, these compul-sory wisdoms can cause misunderstandings, lead-ing to errors. Therefore, it is useful, in orderto avoid unnecessary stumbling during calcula-tions, to introduce separate notation for half-raremarginal probabilities events from s.h-r.eβs. X andits set-phenomena, that is, probabilities that are notgreater than half, in order to distinguish them from free marginal probabilities , to the values of whichthere are no restrictions.So, we will talk about half-rare variables (h-r vari-ables) and free variables , assigning special notationto them : Λ π = { π π₯ , π₯ β X } β [0 , / β X β X -set of half-rare variables , Λ π ( π | π// X ) β [0 , / β π β (1 / , β X β π β π -renumbering Λ π, Λ π€ = { π€ π₯ , π₯ β X } β [0 , β X β X -set of free variables , Λ π€ ( π | π// X ) β [0 , β X β π -renumbering Λ π€, (2.28) Just remember [6], that the formula of π -renumbering any X -set of probabilities of events has the form for π β X : Λ π ( π | π// X ) = { π π₯ , π₯ β π } + { β π π₯ , π₯ β X β π } . HE XIV FAMEMSβ2015 C
ONFERENCE and always interpreting them as probabilities ofevents. In particular, for the half-rare doublet X = { π₯, π¦ } we have: Λ π = { π π₯ , π π¦ } β [0 , / π₯ β [0 , / π¦ β X -set of half-rare variables ,π π₯π¦ ( π π₯ , π π¦ ) β [0 , min { π π₯ , π π¦ } ] β half-rare function of half-rare variables , Λ π€ = { π€ π₯ , π€ π¦ } β [0 , π₯ β [0 , π¦ β X -set of free variables ,π€ π₯π¦ ( π€ π₯ , π€ π¦ ) β [0 , min { π€ π₯ , π€ π¦ } ] β free function of free variables . (2.29) Note 3 (phenomenon replacement between half-rare andfree variables).
For every π β X phenomenonreplacement of half-rare variables Λ π β [0 , / β X byfree variables Λ π€ β [0 , β X and vise-versa is deο¬nedfor π β X by mutually inverse formulas of theset-phenomenon transformation of the form: Λ π = Λ π ( π | X // X ) == β§βͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβ© Λ π€ ( π |β // X ) , π€ π₯ > / , π₯ β X ,. . . , . . . Λ π€ ( π | π// X ) , π€ π₯ (cid:54) / , π₯ β π,π€ π₯ > / , π₯ β X β π,. . . , . . . Λ π€ ( π | X // X ) , π€ π₯ (cid:54) / , π₯ β X (2.30)where for π₯ β X the following agreement is alwaysaccepted (see, for example, paragraph 11.7): π π₯ = {οΈ π€ π₯ , π€ π₯ (cid:54) / , β π€ π₯ , π€ π₯ > / . (2.31) Letβs construct the -Kopula K β Ξ¨ X of a familyof e.p.d.βs of monoplet of events X = { π₯ } with thee.p.d. of the 1st kind {οΈ π ( π// { π₯ } ) , π β { π₯ } }οΈ = { π ( β // { π₯ } ) , π ( π₯// { π₯ } ) } and { π₯ } -monoplet of marginal probabilities { π π₯ } ,where π π₯ = P ( π₯ ) = π ( π₯// { π₯ } ) . In other words, letβs construct a 1-function on theunit X -segment, i.e., a such nonnegative boundednumerical function K : [0 , β [0 , , that for π₯ β X βοΈ π₯ β π β X K (οΈ Λ π€ ( π | π// X ) )οΈ = π€ π₯ . Since for π β X = { π₯ } K (οΈ Λ π€ ( π | π// { π₯ } ) )οΈ = {οΈ K (1 β π€ π₯ ) , π = β , K ( π€ π₯ ) , π = { π₯ } , then a marginal and global normalization of thefunction K are written as: K ( π€ π₯ ) = π€ π₯ , K ( π€ π₯ ) + K (1 β π€ π₯ ) = 1 , (2.32)and the global normalization obviously followsfrom the marginal one, which agrees with Lemma2; and from the marginal normalization it followsthat the 1-copula K of an arbitrary monoplet ofevents X = { π₯ } is deο¬ned for free variables Λ π€ = { π€ π₯ } β [0 , π₯ by a one formula: K ( Λ π€ ) = K ( π€ π₯ ) = π€ π₯ , (2.33)which provides two values on each ( π | Λ π€ ) -penomenon-dom by βfreeβ formulas: K (οΈ Λ π€ ( π | π// { π₯ } ) )οΈ == {οΈ K (1 β π€ π₯ ) = 1 β π€ π₯ , π = β , K ( π€ π₯ ) = π€ π₯ , π = { π₯ } . (2.34)and the e.p.d. of the 1st kind of this monoplet with { π₯ } -monoiplet of probabilities of events Λ π = { π π₯ } β [0 , π₯ are deο¬ned for half-rare variables by the 1-Kopula (2.33) for π β { π₯ } by exactly the same βhalf-rareβ formulas: π ( π// { π₯ } ) = K (οΈ Λ π ( π | π// { π₯ } ) )οΈ == {οΈ K (1 β π π₯ ) = 1 β π π₯ , π = β , K ( π π₯ ) = π π₯ , π = { π₯ } . (2.35) Letβs construct an example of K β Ξ¨ X offamilies of a doublet of events X = { π₯, π¦ } , in otherwords, letβs construct on the unit { π₯, π¦ } -square thesuch nonnegative bounded numerical functions K : [0 , β{ π₯,π¦ } β [0 , , that for all π§ β { π₯, π¦ } βοΈ π§ β π β{ π₯,π¦ } K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ = π€ π§ . Since each 2-set of arguments Λ π€ β [0 , π₯ β [0 , π¦ gen-erates { π₯,π¦ } -phenomenon-dom ( π | Λ π€ ) = { Λ π€, Λ π€ ( π |{ π₯ } ) , Λ π€ ( π |{ π¦ } ) , Λ π€ ( π |β ) } , (2.36)composed from forth its set-phenomena Λ π€ = Λ π€ ( π |{ π₯,π¦ } // { π₯,π¦ } ) = { π€ π₯ , π€ π¦ } , Λ π€ ( π |{ π₯ } // { π₯,π¦ } ) = { π€ π₯ , β π€ π¦ } , Λ π€ ( π |{ π₯ } // { π₯,π¦ } ) = { β π€ π₯ , π€ π¦ } , Λ π€ ( π |β // { π₯,π¦ } ) = { β π€ π₯ , β π€ π¦ } , (2.37) OROBYEV then K ( Λ π€ ) = K ( π€ π₯ , π€ π¦ ) , K (οΈ Λ π€ ( π |{ π₯ } // { π₯,π¦ } ) )οΈ = K ( π€ π₯ , β π€ π¦ ) , K (οΈ Λ π€ ( π |{ π¦ } // { π₯,π¦ } ) )οΈ = K (1 β π€ π₯ , π€ π¦ ) K (οΈ Λ π€ ( π |β // { π₯,π¦ } ) )οΈ = K (1 β π€ π₯ , β π€ π¦ ) , and normalizations for every Λ π€ β [0 , β{ π₯,π¦ } arewritten as: K ( π€ π₯ , π€ π¦ ) + K ( π€ π₯ , β π€ π¦ ) = π€ π₯ , K ( π€ π₯ , π€ π¦ ) + K (1 β π€ π₯ , π€ π¦ ) = π€ π¦ , K ( π€ π₯ , π€ π¦ ) + K ( π€ π₯ , β π€ π¦ )++ K (1 β π€ π₯ , π€ π¦ ) + K (1 β π€ π₯ , β π€ π¦ ) = 1 . The e.p.d. of the 1st kind of doublet of events isdeο¬ned by the 2-Kopula for π β { π₯, π¦ } in half-rarevariables by general formulas: π ( π// { π₯, π¦ } ) = K (οΈ Λ π ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβ¨βͺβͺβͺβ© K (1 β π π₯ , β π π¦ ) , π = β , K ( π π₯ , β π π¦ ) , π = { π₯ } . K (1 β π π₯ , π π¦ ) , π = { π¦ } , K ( π π₯ , π π¦ ) , π = { π₯, π¦ } , = β§βͺβͺβͺβ¨βͺβͺβͺβ© β π π₯ β π π¦ + π π₯π¦ (Λ π ) , π = β ,π π₯ β π π₯π¦ (Λ π ) , π = { π₯ } .π π¦ β π π₯π¦ (Λ π ) , π = { π¦ } ,π π₯π¦ (Λ π ) , π = { π₯, π¦ } , (2.38)where π π₯π¦ (Λ π ) is functional parameter that has asense of probability of double intersection.This e.p.d. of the 1st kind of doublet of events inthe free functional parameters and variables (afterreplacement (2.31)) has the form: π ( π// { π₯, π¦ } ) = K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© π€ π₯ + π€ π¦ β π€ π₯π¦ (1 β π€ π₯ , β π€ π¦ ) ,π€ π₯ > / , π€ π¦ > / β π = β ,π€ π₯ β π€ π₯π¦ ( π€ π₯ , β π€ π¦ ) ,π€ π₯ (cid:54) / , π€ π¦ > / β π = { π₯ } ,π€ π¦ β π€ π₯π¦ (1 β π€ π₯ , π€ π¦ ) ,π€ π₯ > / , π€ π¦ (cid:54) / β π = { π¦ } ,π€ π₯π¦ ( π€ π₯ , π€ π¦ ) ,π€ π₯ (cid:54) / , π€ π¦ (cid:54) / β π = { π₯, π¦ } . (2.39) The simplest example of a 1-function on a { π₯, π¦ } -square is the so-called independent 2-Kopula , which for free variables Λ π€ β [0 , β{ π₯,π¦ } is deο¬ned by theformula: K ( Λ π€ ) = π€ π₯ π€ π¦ . (2.40)This provides it on each ( π | Λ π€ ) -phenomenon the fol-lowing four values: K (οΈ Λ π€ ( π |{ π₯,π¦ } // { π₯,π¦ } ) )οΈ = π€ π₯ π€ π¦ , K (οΈ Λ π€ ( π |{ π₯ } // { π₯,π¦ } ) )οΈ = π€ π₯ (1 β π€ π¦ ) , K (οΈ Λ π€ ( π |{ π¦ } // { π₯,π¦ } ) )οΈ = (1 β π€ π₯ ) π€ π¦ , K (οΈ Λ π€ ( π |β // { π₯,π¦ } ) )οΈ = (1 β π€ π₯ )(1 β π€ π¦ ) . (2.41)Indeed, the so-deο¬ned independent 2-Kopula is a 1-function because βοΈ π₯ β π β{ π₯,π¦ } K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ = π€ π₯ π€ π¦ + π€ π₯ (1 β π€ π¦ ) = π€ π₯ , βοΈ π¦ β π β{ π₯,π¦ } K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ = π€ π₯ π€ π¦ + (1 β π€ π₯ ) π€ π¦ = π€ π¦ . The e.p.d. of the 1st kind of doublet of independentevents with the { π₯, π¦ } -set of probabilities of events Λ π is deο¬ned by four values of the independent 2-Kopula (2.40) on its ( π | Λ π ) -penomenon-dom by gen-eral formulas in half-rare variables (see Fig. 2), i.e.,for π β { π₯, π¦ } : π ( π// { π₯, π¦ } ) = K (οΈ Λ π ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβ¨βͺβͺβͺβ© (1 β π π₯ )(1 β π π¦ ) , π = β ,π π₯ (1 β π π¦ ) , π = { π₯ } . (1 β π π₯ ) π π¦ , π = { π¦ } ,π π₯ π π¦ , π = { π₯, π¦ } . (2.42) With the phenomenal substitution (2.30) half-rare2-Kopula as a function of the free variables takesthe equivalent form: K ( Λ π€ ) = β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© π€ π₯π¦ (οΈ Λ π€ ( π | X // X ) )οΈ , Λ π€ β [0 , / π₯ β [0 , / π¦ ,π€ π₯ β π€ π₯π¦ (οΈ Λ π€ ( π |{ π₯ } // X ) )οΈ , Λ π€ β [0 , / π₯ β (1 / , π¦ ,π€ π¦ β π€ π₯π¦ (οΈ Λ π€ ( π |{ π¦ } // X ) )οΈ , Λ π€ β (1 / , π₯ β [0 , / π¦ ,π€ π₯ + π€ π¦ β π€ π₯π¦ (οΈ Λ π€ ( π |β // X ) )οΈ , Λ π€ β (1 / , π₯ β (1 / , π¦ . (2.43)We rewrite this formula a pair of times, in order tounderstand the properties of the phenomenon sub-stitution of variables and not get confused in the HE XIV FAMEMSβ2015 C
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Figure 2: Graphs of Cartesian representation of the 2-Kopula of a familyof e.p.d.βs of an independent half-rare doublet of events { π₯, π¦ } ; probabil-ities of the 1st kind are marked by different colors : π ( π₯π¦ ) (aqua), π ( π₯ ) (lime), π ( π¦ ) (yellow) π ( β ) (fuchsia). calculations: K ( Λ π€ ) = β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© π€ π₯π¦ ( π€ π₯ , π€ π¦ ) , Λ π€ β [0 , / π₯ β [0 , / π¦ ,π€ π₯ β π€ π₯π¦ ( π€ π₯ , β π€ π¦ ) , Λ π€ β [0 , / π₯ β (1 / , π¦ ,π€ π¦ β π€ π₯π¦ (1 β π€ π₯ , π€ π¦ ) , Λ π€ β (1 / , π₯ β [0 , / π¦ ,π€ π₯ + π€ π¦ β π€ π₯π¦ (1 β π€ π₯ , β π€ π¦ ) , Λ π€ β (1 / , π₯ β (1 / , π¦ ; (2.44)and again with restrictions in the form of the famil-iar βhumanβ inequalities : K ( Λ π€ ) = β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© π€ π₯π¦ ( π€ π₯ , π€ π¦ ) ,π€ π₯ (cid:54) / , π€ π¦ (cid:54) / ,π€ π₯ β π€ π₯π¦ ( π€ π₯ , β π€ π¦ ) ,π€ π₯ (cid:54) / , / < π€ π¦ ,π€ π¦ β π€ π₯π¦ (1 β π€ π₯ , π€ π¦ ) , / < π€ π₯ , π€ π¦ (cid:54) / ,π€ π₯ + π€ π¦ β π€ π₯π¦ (1 β π€ π₯ , β π€ π¦ ) , / < π€ π₯ , / < π€ π¦ (2.45) probabilistic normalization: β (cid:54) ... (cid:54) β is assumed by de-fault. where (cid:54) π€ π₯π¦ ( π€ π₯ , π€ π¦ ) (cid:54) min { π€ π₯ , π€ π¦ } , (cid:54) π€ π₯π¦ ( π€ π₯ , β π€ π¦ ) (cid:54) min { π€ π₯ , β π€ π¦ } , (cid:54) π€ π₯π¦ (1 β π€ π₯ , π€ π¦ ) (cid:54) min { β π€ π₯ , π€ π¦ } , (cid:54) π€ π₯π¦ (1 β π€ π₯ , β π€ π¦ ) (cid:54) min { β π€ π₯ , β π€ π¦ } (2.46)are the FrΒ΄echet inequalities for π€ π₯π¦ as the half-rare probability of double intersection of half-rareevents: either half-rare events π₯ or π¦ , or their half-rare complements, when events π₯ or π¦ are not half-rare.Note that the conditional formulas (2.43), (2.44) and(2.45) can not be rewritten as four unconditionalformulas, because these conditions are in the right,and not in the left. This is explained exclusively bythe properties of the phenomenon replacement ofhalf-rare variables by free ones (2.30), which, forthis reason, leads to formulas that are convenientfor calculations. Note 4 (half-rare 2-Kopula of free variables of anindependent doublet of events).
With the functionalparameter π€ π₯π¦ ( π€ π₯ , π€ π¦ ) = π€ π₯ π€ π¦ , which corre-sponds to the probability of double intersectionof independent events π₯ and π¦ happening withprobabilities π€ π₯ and π€ π¦ , and means, of course, thatthe all the following four equations are satisο¬ed: π€ π₯π¦ ( π€ π₯ , π€ π¦ ) = π€ π₯ π€ π¦ ,π€ π₯π¦ ( π€ π₯ , β π€ π¦ ) = π€ π₯ (1 β π€ π¦ ) ,π€ π₯π¦ (1 β π€ π₯ , π€ π¦ ) = (1 β π€ π₯ ) π€ π¦ ,π€ π₯π¦ (1 β π€ π₯ , β π€ π¦ ) = (1 β π€ π₯ )(1 β π€ π¦ ); from (2.45) it follows that a half-rare 2-Kopula fromfree variables of the family of e.p.d.βs of the inde-pendent doublet of events X = { π₯, π¦ } with X -sets offree marginal probabilities Λ π€ = { π€ π₯ , π€ π¦ } β [0 , β X has the same view on all ( π | Λ π€ ) -phenomenon-doms: K ( Λ π€ ) = π€ π₯ π€ π¦ . (2.47) An example of a 1-function on a { π₯, π¦ } -square is theso-called upper 2-Kopula of FrΒ΄echet , which suggeststhe probabilities of a double intersection to be itsupper FrΒ΄echet boundary. In other words, the onlyfunctional free parameter in (2.39) is: π€ π₯π¦ ( Λ π€ ) = π€ + π₯π¦ ( Λ π€ ) = min { π€ π₯ , π€ π¦ } . (2.48) OROBYEV The upper 2-Kopula of FrΒ΄echet from free variables Λ π€ β [0 , β{ π₯,π¦ } is deο¬ned by the formulas: π ( π// { π₯, π¦ } ) = K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© π€ π₯ + π€ π¦ β { β π€ π₯ , β π€ π¦ } ,π€ π₯ > / , π€ π¦ > / β π = β ,π€ π₯ β min { π€ π₯ , β π€ π¦ } ,π€ π₯ (cid:54) / , π€ π¦ > / β π = { π₯ } ,π€ π¦ β min { β π€ π₯ , π€ π¦ } ,π€ π₯ > / , π€ π¦ (cid:54) / β π = { π¦ } , min { π€ π₯ , π€ π¦ } ,π€ π₯ (cid:54) / , π€ π¦ (cid:54) / β π = { π₯, π¦ } . (2.49)After simple transformations, these formulas pro-vide the upper 2-Kopula of FrΒ΄echet on each ( π | Λ π€ ) -phenomenon-dom the following four values of freevariables: π ( π// { π₯, π¦ } ) = K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© min { π€ π₯ , π€ π¦ } ,π€ π₯ > / , π€ π¦ > / β π = β , max { , π€ π₯ + π€ π¦ β } ,π€ π₯ (cid:54) / , π€ π¦ > / β π = { π₯ } , max { , π€ π₯ + π€ π¦ β } ,π€ π₯ > / , π€ π¦ (cid:54) / β π = { π¦ } , min { π€ π₯ , π€ π¦ } ,π€ π₯ (cid:54) / , π€ π¦ (cid:54) / β π = { π₯, π¦ } , = β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© π€ + π₯π¦ ( π€ π₯ , π€ π¦ ) ,π€ π₯ > / , π€ π¦ > / β π = β ,π€ β π₯π¦ ( π€ π₯ , π€ π¦ ) ,π€ π₯ (cid:54) / , π€ π¦ > / β π = { π₯ } ,π€ β π₯π¦ ( π€ π₯ , π€ π¦ ) ,π€ π₯ > / , π€ π¦ (cid:54) / β π = { π¦ } ,π€ + π₯π¦ ( π€ π₯ , π€ π¦ ) ,π€ π₯ (cid:54) / , π€ π¦ (cid:54) / β π = { π₯, π¦ } , (2.50)which, as can be seen, are deο¬ned by the upper andlower FrΒ΄echet boundaries of the probability of dou-ble intersecyion, depending on the combination ofthe values of the free variables.The same four values from the half-rare variableshave the form: π ( π// { π₯, π¦ } ) = K (οΈ Λ π ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβ© min { β π π₯ , β π π¦ } , π = β , max { , π π₯ β π π¦ } , π = { π₯ } , max { , π π¦ β π π₯ } , π = { π¦ } , min { π π₯ , π π¦ } , π = { π₯, π¦ } . (2.51) If π π₯ (cid:62) π π¦ , this formula takes the form: π ( π// { π₯, π¦ } ) = K (οΈ Λ π ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβ© β π π₯ , π = β ,π π₯ β π π¦ , π = { π₯ } , , π = { π¦ } ,π π¦ , π = { π₯, π¦ } . (2.52)And if π π₯ < π π¦ , then this formula takes the form: π ( π// { π₯, π¦ } ) = K (οΈ Λ π ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβ© β π π¦ , π = β , , π = { π₯ } ,π π¦ β π π₯ , π = { π¦ } ,π π¦ , π = { π₯, π¦ } . (2.53)So the deο¬nite upper 2-Kopula of FrΒ΄echet is indeeda 1-function, due to the fact that when π€ π₯ (cid:62) π€ π¦ βοΈ π₯ β π β{ π₯,π¦ } K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ = π€ π¦ + ( π€ π₯ β π€ π¦ ) = π€ π₯ , βοΈ π¦ β π β{ π₯,π¦ } K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ = π€ π¦ + 0 = π€ π¦ , (2.54)and when π€ π₯ < π€ π¦ βοΈ π₯ β π β{ π₯,π¦ } K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ = π€ π₯ + 0 = π€ π₯ , βοΈ π¦ β π β{ π₯,π¦ } K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ = π€ π₯ + ( π€ π¦ β π€ π₯ ) = π€ π¦ . (2.55) A once more example of a 1-function on a { π₯, π¦ } -square is the so-called lower 2-Kopula of FrΒ΄echet ,which suggests the probabilities of a double inter-section to be its upper FrΒ΄echet boundary. In otherwords, the only functional free parameter in (2.39)is: π€ π₯π¦ ( Λ π€ ) = π€ β π₯π¦ ( Λ π€ ) = max { , π€ π₯ + π€ π¦ β } . (2.56) HE XIV FAMEMSβ2015 C
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Figure 3: Graphs of the Cartesian representation of the upper 2-Kopula ofFrΒ΄echet of a family of e.p.d.βs of half-rare doublet of events { π₯, π¦ } ; proba-bilities of the 1st kind are marked by different colors: π ( π₯π¦ ) (aqua), π ( π₯ ) (lime), π ( π¦ ) (yellow) π ( β ) (fuchsia).Figure 4: Graphs of the Cartesian representation of the lower 2-Kopula ofFrΒ΄echet of a family of e.p.d.βs of half-rare doublet of events { π₯, π¦ } ; proba-bilities of the 1st kind are marked by different colors: π ( π₯π¦ ) (aqua), π ( π₯ ) (lime), π ( π¦ ) (yellow) π ( β ) (fuchsia). The lower 2-Kopula of FrΒ΄echet from free variables Λ π€ β [0 , β{ π₯,π¦ } is deο¬ned by the formulas: π ( π// { π₯, π¦ } ) = K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© π€ π₯ + π€ π¦ β { , β π€ π₯ β π€ π¦ } ,π€ π₯ > / , π€ π¦ > / β π = β ,π€ π₯ β max { , π€ π₯ β π€ π¦ } ,π€ π₯ (cid:54) / , π€ π¦ > / β π = { π₯ } ,π€ π¦ β max { , π€ π¦ β π€ π₯ } ,π€ π₯ > / , π€ π¦ (cid:54) / β π = { π¦ } , max { , π€ π₯ + π€ π¦ β } ,π€ π₯ (cid:54) / , π€ π¦ (cid:54) / β π = { π₯, π¦ } . (2.57) After simple transformations, these formulas pro-vide the lower 2-Kopula of FrΒ΄echet on each ( π | Λ π€ ) -phenomenon-dom the following four values of freevariables: π ( π// { π₯, π¦ } ) = K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© max { , π€ π₯ + π€ π¦ β } ,π€ π₯ > / , π€ π¦ > / β π = β , min { π€ π₯ , π€ π¦ } ,π€ π₯ (cid:54) / , π€ π¦ > / β π = { π₯ } , min { π€ π₯ , π€ π¦ } ,π€ π₯ > / , π€ π¦ (cid:54) / β π = { π¦ } , max { , π€ π₯ + π€ π¦ β } ,π€ π₯ (cid:54) / , π€ π¦ (cid:54) / β π = { π₯, π¦ } , = β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© π€ β π₯π¦ ( π€ π₯ , π€ π¦ ) ,π€ π₯ > / , π€ π¦ > / β π = β ,π€ + π₯π¦ ( π€ π₯ , π€ π¦ ) ,π€ π₯ (cid:54) / , π€ π¦ > / β π = { π₯ } ,π€ + π₯π¦ ( π€ π₯ , π€ π¦ ) ,π€ π₯ > / , π€ π¦ (cid:54) / β π = { π¦ } ,π€ β π₯π¦ ( π€ π₯ , π€ π¦ ) ,π€ π₯ (cid:54) / , π€ π¦ (cid:54) / β π = { π₯, π¦ } , (2.58)which, as can be seen, are also deο¬ned by the upperand lower FrΒ΄echet boundaries of the probability ofdouble intersection only in other combinations ofthe values of the free variables.The same four values from the half-rare variableshave the more simple form: π ( π// { π₯, π¦ } ) = K (οΈ Λ π ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβ© max { , β π π₯ β π π¦ } , π = β , min { π π₯ , β π π¦ } , π = { π₯ } , min { β π π₯ , π π¦ } , π = { π¦ } , max { , π π₯ + π π¦ β } , π = { π₯, π¦ } , = β§βͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβ© β π π₯ β π π¦ , π = β ,π π₯ , π = { π₯ } ,π π¦ , π = { π¦ } , , π = { π₯, π¦ } , (2.59)So the deο¬nite lower 2-Kopula of FrΒ΄echet is indeeda 1-function, due to the fact that for all half-rarevariables βοΈ π₯ β π β{ π₯,π¦ } K (οΈ Λ π ( π | π// { π₯,π¦ } ) )οΈ = π π₯ + 0 = π π₯ , βοΈ π¦ β π β{ π₯,π¦ } K (οΈ Λ π€ ( π | π// { π₯,π¦ } ) )οΈ = π π¦ + 0 = π π¦ . (2.60) OROBYEV { π₯, π¦ } with functional weight parameter (2.63) inthe formula (2.61); probabilities of the 1st kind are marked by differentcolors: π ( π₯π¦ ) (aqua), π ( π₯ ) (lime), π ( π¦ ) (yellow) π ( β ) (fuchsia). A rather general example of a 1-function on a { π₯, π¦ } -square is the convex combinations of the upper,lower, and independent 2-Kopulas ofwise FrΒ΄echet ,which propose the probabilities of a pair inter-section to become a convex combination (this isallowed by the lemma 3) of its upper and lowerFrΒ΄echet boundaries, as well as the probability ofdouble intersection of independent events withsome functional weighting parameter. A convex combination of the lower and upper 2-Kopula of FrΒ΄echet can be ensured by the uniquefunctional free parameter π€ π₯π¦ ( Λ π€ ) in (2.39), in whichthe probability of double intersection is computedby the following formula: π€ π₯π¦ ( Λ π€ ) == (1 β πΌ ) / π€ β π₯π¦ ( Λ π€ ) + (1+ πΌ ) / π€ + π₯π¦ ( Λ π€ ) (2.61)where πΌ = πΌ ( Λ π€ ) β [ β , is an arbitrary function on [0 , β{ π₯,π¦ } with values from [ β , , and π€ β π₯π¦ ( Λ π€ ) = max { , π€ π₯ + π€ π¦ β } ,π€ + π₯π¦ ( Λ π€ ) = min { Λ π€ } (2.62) are the lower and upper FrΒ΄echet-boundaries ofprobability of double intersection.For πΌ = β , the probability π€ π₯π¦ ( Λ π€ ) = π€ β π₯π¦ ( Λ π€ ) co-incides with the lower FrΒ΄echet boundary of half-rare marginal probabilities; for πΌ = 1 , the prob-ability π€ π₯π¦ ( Λ π€ ) = π€ + π₯π¦ ( Λ π€ ) coincides with the upperFrΒ΄echet boundary of marginal probabilities. Un-fortunately, these are the properties of a convexcombination such that for πΌ = 0 the probability ofdouble intersection is equal to half of the sum ofits lower and upper FrΒ΄echet boundaries: π€ π₯π¦ ( Λ π€ ) =( π€ β π₯π¦ ( Λ π€ ) + π€ + π₯π¦ ( Λ π€ )) / (see Figure6), and not an inde-pendent 2-Kopula, no matter how much we wantit. This βblunderβ of the convex combination caneasily be corrected by conjugation of two convexcombinations, as done below.In Fig. 5 it is a graph of this 2-Kopula for a delib-erately intricate weight function with values from [ β , : πΌ = πΌ ( Λ π€ ) = sin(15( π€ π₯ β π€ π¦ )) . (2.63) Figure 6: Graphs of the Cartesian representation of the convex combi-nation of upper and lower 2-Kopulas of FrΒ΄echet of a family of e.p.d.βs ofhalf-rare doublet of events { π₯, π¦ } with the constant functional weight pa-rameter πΌ ( Λ π€ ) = 0 in the formula (2.61); probabilities of the 1st kind aremarked by different colors: π ( π₯π¦ ) (aqua), π ( π₯ ) (lime), π ( π¦ ) (yellow) π ( β ) (fuchsia). We construct two convex combinations of the theindependent 2-copula and the lower and upper 2-Kopulas of FrΒ΄echet. The conjugation of these two HE XIV FAMEMSβ2015 C
ONFERENCE convex combinations can be ensured by the uniquefunctional free parameter π€ π₯π¦ ( Λ π€ ) in (2.39) by thefollowing conjugation formula for two convex com-binations: π€ π₯π¦ ( Λ π€ ) == β§βͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβ© π€ + π₯π¦ ( Λ π€ ) max { Λ π€ } (1+ πΌ ) ,πΌ (cid:54) π€ + π₯π¦ ( Λ π€ ) (οΈ max { Λ π€ } (1 β πΌ )+ πΌ )οΈ ,πΌ > , (2.64)where πΌ = πΌ ( Λ π€ ) β [ β , is an arbitrary function on [0 , β{ π₯,π¦ } with values from [ β , , and π€ + π₯π¦ ( Λ π€ ) = min { Λ π€ } (2.65)is the upper FrΒ΄echet-boundary of probability ofdouble intersection. For πΌ = 0 , the probability π€ π₯π¦ ( Λ π€ ) = min { Λ π€ } max { Λ π€ } = π€ π₯ π€ π¦ coincides with theprobability of double intersection of independentevents; for πΌ = β , the probability π€ π₯π¦ ( Λ π€ ) = 0 coin-cides with the lower FrΒ΄echet-boundary of half-raremarginal probabilities; for πΌ = 1 , the probability π€ π₯π¦ ( Λ π€ ) = π€ + π₯π¦ ( Λ π€ ) coincides with the upper FrΒ΄echet-boundary of marginal probabilities.In Fig. 5 it is a graph of this 2-Kopula for the sameweight function with values from [ β , as in theprevious example. πΌ = πΌ ( Λ π€ ) = sin(15( π€ π₯ β π€ π¦ )) . (2.66) Definition 6 (inserted s.e.βs).
For each pais ofs.e.βs π³ and π΄ with the joint e.p.d. { π ( π + π // π³ + π΄ ) } , π β π³ , π β π΄} (3.1)for every π β π΄ the π -inserted s.e.βs, generatedby π³ , in the frame s.e. π΄ are s.e.βs, which are de-noted by π³ ( β© π// π΄ ) , and deο¬ned as the following M-intersection : π³ ( β© π// π΄ ) = π³ ( β© ) { ter ( π // π΄ ) } == { π₯ β© ter ( π // π΄ ) , π₯ β π³ } (3.2)and have the e.p.d., which coincides with the pro-jection of the joint e.p.d. (3.1) for ο¬xed π β π΄ andevery π β π³ : π ( π ( β© ) { ter ( π // π΄ ) } ) = π ( π + π // π³ + π΄ ) } . (3.3) M-intersection is an intersection by Minkowski.
Definition 7 (event-probabilistic pseudo-distribution of an inserted s.e.).
For each π β π΄ the π -inserted s.e. π³ ( β© π// π΄ ) = π³ ( β© ) { ter ( π // π΄ ) } (3.4)with the e.p.d. (3.3) has the event-probabilistic π -pseudo-distribution , which is deο¬ned as a set ofprobabilitieso of terraced events that coincide withprobabilities from the e.p.d. (3.3) for all π β π³ ex-cepting π = β : π ( π ) ( π + π // π³ + π΄ ) == {οΈ π ( π + π // π³ + π΄ ) , π ΜΈ = β ,π ( π // π³ + π΄ ) β π ( π // π΄ ) , π = β , = β§βͺβ¨βͺβ© π ( π + π // π³ + π΄ ) , π ΜΈ = β ,π ( π // π΄ ) β βοΈ π ΜΈ = β π βπ³ π ( π + π // π³ + π΄ ) , π = β . (3.5)The sum of all probabilities from every π -pseudo-distribution (3.5) is π ( π // π΄ ) = P ( ter ( π // π΄ )) , theprobability of a terraced event, generated by the frame s.e. π΄ , in which the given s.e. π³ ( β© π// π΄ ) is in-serted.Thus, the only difference of e.p.d.βs of π -inserteds.e.βs from their event-probabilistic π -pseudo-distributions, lies in the fact that the sums of theprobabilities of the terraced events, from which the π -pseudo-distributions are composed, are normal-ized not by unity, but by the probabilities of the cor-responding frame terraces events π ( π // π΄ ) . And thesum of the normalizing constants by π β π΄ is obvi-ously equal to one. Note 5 (symmetry of inserted and frame s.e.βs).
InDeο¬nition 6 the s.e. π³ and π΄ can always beswapped, i.e., to take the s.e. π³ on a role of the frame one, and the s.e. π΄ to take on a role of s.e.,that generates π -inserted s.e.βs for every π β π³ : π΄ ( β© π// π³ ) = π΄ ( β© ) { ter ( π// π³ ) } == { π¦ β© ter ( π// π³ ) , π¦ β π΄} . (3.6) Note 6 (M-sum of the all inserted s.e.βs).
The M-sum of the all π -inserted s.e.βs π³ ( β© π// π΄ ) for π β π΄ forms the given s.e. π³ : π³ = (οΈβοΈ)οΈ π βπ΄ π³ ( β© π// π΄ ) == π³ ( β©β // π΄ ) (+) ... (+) π³ ( β©π΄ // π΄ ) β β |π΄| . (3.7) Note 7 (charcterization of π -inserted s.e.βs by con-ditional e.p.d.βs of the 1st kind). The e.p.d. of π -inserted s.e. π³ ( β© π// π΄ ) with every π β π΄ has a M-sum is a sum by Minkowski.
OROBYEV form for π β π³ : π (οΈ π ( β© ) { ter ( π // π΄ ) } // π³ ( β© π// π΄ ) )οΈ == {οΈ π ( π + π // π³ + π΄ ) , β ΜΈ = π β π³ ,π ( π // π³ + π΄ ) + 1 β π ( π // π΄ ) , π = β , = {οΈ π ( π//
π³ |
π // π΄ ) π ( π // π΄ ) , β ΜΈ = π β π³ , β (1 β π ( β // π³ |
π // π΄ )) π ( π // π΄ ) , π = β (3.8)where for every π β π΄ π ( π//
π³ |
π // π΄ ) = π ( π + π // π³ + π΄ ) π ( π // π΄ ) , (3.9)i.e., the probabilities of the 1st kind, forming for π β π³ the π -conditional e.p.d. of the 1st kind ofthe s.e. π³ with respect to terraced event ter ( π // π΄ ) generated by the s.e. π΄ .In other words, π -inserted s.e. π³ ( β© π// π΄ ) for π βπ΄ are characterized by formulas (3.8) and π -conditional e.p.d.βs of the 1st kind of the s.e. π³ withrespect to the terraced event ter ( π // π΄ ) , generatedby the s.e. π΄ . Note 8 (mutual characterization of conditionale.p.d.βs of the 1st kind and pseudo-distributions ofinserted s.e.βs).
The connection between each π -pseudo-distribution of the π -inserted s.e. with thecorresponding π -conditional e.p.d. looks simpler.It is suο¬cient for each ο¬xed π β π΄ to normalizeall its probabilities of βinsertedβ terraced events bythe probability of a terraced event π ( π // π΄ ) in orderto obtain corresponding to the π -conditional prob-abilities regarding the fact that the correspondingframe terraces event ter ( π // π΄ ) happened. As aresult, we have the following obvious inversionformulas: π ( π//
π³ |
π // π΄ ) = π ( π ) ( π + π // π³ + π΄ ) π ( π // π΄ ) ,π ( π ) ( π + π // π³ + π΄ ) = π ( π//
π³ |
π // π΄ ) π ( π // π΄ ) . (3.10) Note 9 (about the appropriateness of the conceptof inserted s.e.βs).
It would seem, why developa theory of inserted s.e.βs, pseudo-distributionsof which are simply characterized by conditionale.p.d.βs. Is not it better to instead practice thetheory of conditional e.p.d., especially since thistheory has long had excellent recommendationsin many areas. However, in eventology, as thetheory of events, which prefers to work directlywith sets of events, there is one rather seriousobjection. The fact is that conditional e.p.d., asany e.p.d. in eventology, there must be a set ofsome events, in this case, a set of well-deο¬nedβconditional eventsβ. But until now it has notbeen possible to give a satisfactory deο¬nition ofthe βconditional eventβ, except for my impracticaldeο¬nition in [7]. So, the inserted s.e.βs are a com-pletely satisfactory βsurrogateβ deο¬nition of thesets of βconditional eventsβ. Such that e.p.d.βs of inserted s.e.βs, although they do not coincide withthe desired conditional e.p.d.βs, but are associatedwith them by well-deο¬ned mutual-inverse trans-formations. As a result, inserted s.e.βs play the roleof a convenient eventological tool for working withconditional e.p.d.βs of a one set of events regardingterrace events generated by another set of events.
Example 1 (two inserted s.e.βs in a frame monoplet).
Let in formulas (3.1) the s.e. π³ is an arbitrary set,and the s.e. π΄ = { π¦ } is a frame monoplet of events,which have the joint e.p.d. in a form: { π ( π + π // π³ + { π¦ } ) , π β π³ , π β { π¦ }} . (3.11)Then there is the { π¦ } -inserted s.e. and the β -inserted s.e.: π³ ( β©{ π¦ } // { π¦ } ) = { π₯ β© π¦, π₯ β π³ } , π³ ( β©β // { π¦ } ) = { π₯ β© π¦ π , π₯ β π³ } . (3.12)These inserted s.e.βs are characterized for every oftwo subsets of the monoplet π = { π¦ } β { π¦ } and π = β β { π¦ } by formulas (3.3) and by two correspondinge.p.d.βs { π ( π// π³ + { π¦ } ) } , π β π³ } , { π ( π + { π¦ } // π³ + { π¦ } ) } , π β π³ } . (3.13)which by formulas (3.5) deο¬ne two π -pseudo-distributions for π β π³ : π ( { π¦ } ) ( π + { π¦ } // π³ + { π¦ } ) == {οΈ π ( π + { π¦ } // π³ + { π¦ } ) , π ΜΈ = β ,π ( { π¦ } // π³ + { π¦ } ) β π ( { π¦ } // { π¦ } ) ,π = β , = {οΈ π ( π + { π¦ } // π³ + { π¦ } ) , π ΜΈ = β ,π ( { π¦ } // π³ + { π¦ } ) β π π¦ , π = β . (3.14) π ( β ) ( π// π³ + { π¦ } ) == {οΈ π ( π// π³ + { π¦ } ) , π ΜΈ = β ,π ( β // π³ + { π¦ } ) β π ( β // { π¦ } ) , π = β , = {οΈ π ( π// π³ + { π¦ } ) , π ΜΈ = β ,π ( β // π³ + { π¦ } ) β π π¦ , π = β , (3.15)where π π¦ = P ( π¦ ) is a probability of the frame event π¦ β { π¦ } .First of all, note that the sum of the probabilitiesof terraced events from the { π¦ } -pseudo-distribution(3.14) is π π¦ , and the probabilities of the β -pseudo-distribution (3.15) is β π π¦ ; and secondly, that thesetwo pseudo-distributions deο¬ne a joint e.p.d. of thes.e. π³ and the monoplet { π¦ } , i.e., e.p.d. of the s.e. π³ + { π¦ } , which is related to them by fairly obviousformulas for π β π³ + { π¦ } : π ( π// π³ + { π¦ } ) = {οΈ π ( { π¦ } ) ( π// π³ + { π¦ } ) , π¦ β π,π ( β ) ( π// π³ + { π¦ } ) , π¦ ΜΈβ π. (3.16) HE XIV FAMEMSβ2015 C
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The formulas (3.16) are recurrent, connecting thee.p.d. of s.e. π³ + { π¦ } with two pseudo-distributionsof the inserted s.e. π³ β² = π³ ( β©{ π¦ } // π΄ ) and π³ β²β² = π³ ( β©β // π΄ ) whose power is less by one. The inversionformulas (3.10) allow recurrence formulas (3.16) toexpress the e.p.d. of π³ + { π¦ } via the conditionale.p.d. with respect to one of its events π¦ β π³ + { π¦ } and its complements π¦ π = β¦ β π¦ : π ( π// π³ + { π¦ } ) == β§βͺβͺβͺβ¨βͺβͺβͺβ© π ( π//
π³ | { π¦ } // { π¦ } ) π π¦ , π = π + { π¦ } ,π β π³ ; π ( π//
π³ | β // { π¦ } )(1 β π π¦ ) , π = π,π β π³ , (3.17)where π β π³ + { π¦ } . Note that these formulas, like(3.16), can be used recursively to express the e.p.d.of s.e. π³ + { π¦ } through two conditional e.p.d.βs ofthe s.e. π³ whose power is less by one. Definition 8 (inserted Kopulas).
The π -Kopulasof π -inserted π -s.e.βs π³ ( β© π// π΄ ) = π³ ( β© ) { ter ( π // π΄ ) } == { π₯ β© ter ( π // π΄ ) , π₯ β π³ } , (3.18)which for each π β π΄ are deο¬ned (see Deο¬nition6) as intersections by Minkowsi of the s.e. π³ withterraced events ter ( π // π΄ ) , generated by the s.e. π΄ ,are called the π -inserted π -Kopulas with respect tothe s.e. π΄ . Such π -inserted π -Kopulas character-izes e.p.d.βs of the 1st kind of π -inserted π -s.e.βs byformulas for π β π³ π ( π ( β© ) { ter ( π // π΄ ) } ) == K ( π ) (οΈ Λ π ( π | π ( β© π// π΄ ) // π³ ( β© π// π΄ ) ) )οΈ , (3.19)where Λ π ( π |π³ ( β© π// π΄ ) // π³ ( β© π// π΄ ) ) = Λ π ( π ) == {οΈ π ( π ) π₯ , π₯ β π³ }οΈ = {οΈ P ( π₯ β© ter ( π // π΄ )) , π₯ β π³ }οΈ (3.20)is the set of probabilities of βinsertedβ marginalevents from the π³ ( β© π// π΄ ) , and Λ π ( π | π ( β© π// π΄ ) // π³ ( β© π// π΄ ) ) = {οΈ π ( π ) π₯ , π₯ β π }οΈ ++ {οΈ π ( π // π³ ) β π ( π ) π₯ , π₯ β π³ β π }οΈ (3.21)are π -phenomena of the set of βinsertedβ marginalprobabilities Λ π ( π ) .We also need to deο¬ne an inserted π -pseudo-Kopula with respect to the s.e. π΄ , which character-izes the π -pseudo-distribution of the π -inserted s.e. π³ ( β© π// π΄ ) , inserted into the terraces event ter ( π // π΄ ) generated by the s.e. π΄ . Although the π -pseudo-Kopula is not a Kopula, i.e., is not a 1-function, it hasproperties very reminiscent of the Kopula proper-ties. Definition 9 (inserted pseudo-Kopulas).
The π -pseudo-Kopula of the π -pseudo-distribution of π -inserted s.e. π³ ( β© π ) = π³ ( β© π// π΄ ) = π³ ( β© ) { ter ( π // π΄ ) } (3.22)with respect to the s.e. π΄ is a such function K ( π ) on π³ -hypercube with sides [0 , π ( π // π΄ )] that1) is non-negative : K ( π ) (οΈ Λ π€ ( π | π ( β© π ) // X ( β© π ) ) )οΈ (cid:62) (3.23)for Λ π€ ( π | π ( β© π ) // X ( β© π ) ) β [0 , π ( π // π΄ )] βπ³ π β X ;2) satisο¬es the π -marginal equalities for π₯ β X : βοΈ π₯ β π β X K ( π ) (οΈ Λ π€ ( π | π ( β© π ) // X ( β© π ) ) )οΈ = π€ ( π | π ( β© π ) // X ( β© π ) ) π₯ β© ter ( π// π³ ) (3.24)where Λ π€ ( π | π ( β© π ) // π³ ( β© π ) ) = {οΈ π€ ( π | π ( β© π ) // π³ ( β© π ) ) π₯ β© ter ( π// π³ ) , π₯ β π³ }οΈ (3.25)is a π -phenomenon of the π³ -set of marginal prob-abilities of the π -pseudo-distribution of π -inserteds.e. π³ ( β© π ) , i.e., π€ ( π | π ( β© π ) // π³ ( β© π ) ) π₯ β© ter ( π// π³ ) == {οΈ π€ π₯ β© ter ( π// π³ ) , π₯ β π,π ( π // π΄ ) β π€ π₯ β© ter ( π// π³ ) , π₯ β π³ β π. (3.26)From (3.24) and (3.26) it follows the probabilistic π -normalization of pseudo-Kopula : βοΈ π βπ³ K ( π ) (οΈ Λ π€ ( π | π ( β© π ) // π³ ( β© π ) ) )οΈ = π ( π // π΄ ) . (3.27)And from (3.23) and (3.27) it follows the terrace-by-terrace probabilistic π -normalization of pseudo-Kopula : (cid:54) K ( π ) (οΈ Λ π€ ( π | π ( β© π ) // π³ ( β© π ) ) )οΈ (cid:54) π ( π // π΄ ) (3.28)for π β X .Such π -pseudo-Kopulas characterize the π -pseudo-distribution (3.5) of π -inserted s.e.βs π³ ( β© π ) by for-mulas for π β π³ π ( π ) ( π + π // π³ + π΄ ) == K ( π ) (οΈ Λ π ( π | π ( β© π// π΄ ) // π³ ( β© π// π΄ ) ) )οΈ , (3.29) OROBYEV where Λ π ( π |π³ ( β© π// π΄ ) // π³ ( β© π// π΄ ) ) = Λ π ( π ) == {οΈ π ( π ) π₯ , π₯ β π³ }οΈ = {οΈ P ( π₯ β© ter ( π // π΄ )) , π₯ β π³ }οΈ (3.30)is a set of π -marginal probabilities, coinciding withthe set of marginal probabilities of π -inserted s.e.βs π³ ( β© π ) , and Λ π ( π | π ( β© π// π΄ ) // π³ ( β© π// π΄ ) ) = {οΈ π ( π ) π₯ , π₯ β π }οΈ ++ {οΈ π ( π // π³ ) β π ( π ) π₯ , π₯ β π³ β π }οΈ (3.31)are π -phenomena of the set π -marginal probabili-ties Λ π ( π ) . Definition 10 (conditional Kopulas).
The π -Kopulas, characterizing π -inserted e.p.d.βs of the1st kind of the π -s.e. π³ with respect to the terracedevent ter ( π // π΄ ) , generated by the s.e. π΄ , i.e., e.p.d.βsof the 1st kind, deο¬ned by joint e.p.d. π³ and π΄ byformulas with ο¬xed π β π΄ for π β π³ : π ( π//
π³ |
π // π΄ ) = π ( π + π // π³ + π΄ ) π ( π // π΄ ) , (3.32)are called the π -conditional π -Kopulas of the π -s.e. π³ with respect to the terraced event ter ( π // π΄ ) , gen-erated by the s.e. π΄ .Such π -cvonditional π -Kopulas characterize the π -conditional e.p.d. of the 1st kind (3.32) by formulasfor π β π³ : π ( π//
π³ |
π // π΄ ) = K | π (οΈ Λ π ( π | π//
π³ |
π// π΄ ) )οΈ , (3.33)where Λ π ( π |π³ // π³ |
π// π΄ ) = Λ π | π = {οΈ π | ππ₯ , π₯ β π³ }οΈ == {οΈ P ( π₯ β© ter ( π // π΄ )) /π ( π // π΄ ) , π₯ β π³ }οΈ (3.34)is a set of conditional marginal probabilities ofevents π₯ β π³ with respect to the terraced event ter ( π // π΄ ) , and Λ π ( π | π//
π³ |
π// π΄ ) == {οΈ π | ππ₯ , π₯ β π }οΈ + {οΈ β π | ππ₯ , π₯ β π³ β π }οΈ (3.35)are π -phenomenon of the set of conditionalmarginal probabilities Λ π | π . Note 10 (connection between conditional and ββin-sertedββ marginal probaabilities).
Conditionalmarginal probabilities are connected with βin-sertedβ marginal probabilities for π₯ β π³ by theformula of conditional probability: π | ππ₯ = 1 π ( π // π΄ ) π ( π ) π₯ , (3.36)since βinsertedβ marginal probabilities (3.20) areprobabilities of intersections of events π₯ β π³ with the terraced event ter ( π // π΄ ) . The connection be-tween the corresponding set of conditional βin-sertedβ marginal probabilities we shall write in thesimilar way: Λ π | π = 1 π ( π // π΄ ) Λ π ( π ) , Λ π ( π | π//
π³ |
π// π΄ ) = 1 π ( π // π΄ ) Λ π ( π | π ( β© π// π΄ ) // π³ ( β© π// π΄ ) ) . (3.37) Note 11 (connection between conditional Kopulasand inserted pseudo-Kopulas).
From Deο¬nition 10of conditional Kopula and Deο¬nition 9 of insertedpseudo-Kopula with respect to the s.e. π΄ , and alsofrom the formula (3.37) it follows the simple in-version formulas that connect conditional Kopulasand inserted Pseudo-Kopulas of the family of setsof events π³ for π β π³ : K | π (οΈ Λ π ( π | π//
π³ |
π// π΄ ) )οΈ == 1 π ( π // π΄ ) K ( π ) (οΈ π ( π // π΄ )Λ π ( π | π//
π³ |
π// π΄ ) )οΈ , K ( π ) (οΈ Λ π ( π | π ( β© π// π΄ ) // π³ ( β© π// π΄ ) ) )οΈ == π ( π // π΄ ) K | π (οΈ π ( π // π΄ ) Λ π ( π | π ( β© π// π΄ ) // π³ ( β© π// π΄ ) ) )οΈ . (3.38) Note 12 (two formulas of full probability for aKopula).
The Kopula K of s.e. π³ is expressedthrough π -conditional Kpulas K | π for π β π΄ by theusual formula of full probability: K (οΈ Λ π ( π | π// π³ ) )οΈ = βοΈ π βπ΄ K | π (οΈ Λ π ( π | π//
π³ |
π// π΄ ) )οΈ π ( π // π΄ ) . (3.39)From (3.39) and (3.38) we obtain an analogue of theformula of total probability β the representationof the Kopula of s.e. π³ in the form of sum of π -pseudo-Kopulas by π β π΄ : K (οΈ Λ π ( π | π// π³ ) )οΈ = βοΈ π βπ΄ K ( π ) (οΈ Λ π ( π | π ( β© π// π΄ ) // π³ ( β© π// π΄ ) ) )οΈ . (3.40) Note 13 (Kopula of a sum of sets).
A Kopula ofsum π³ + π΄ of two s.e.βs π³ and π΄ characterizes theirjoint e.p.d. of the 1st kind and by deο¬nition has theform π ( π + π // π³ + π΄ ) = K (οΈ Λ π ( π | π + π// π³ + π΄ ) )οΈ , (3.41)where Λ π ( π | π + π// π³ + π΄ ) = { π π₯ , π₯ β π } + { π π¦ , π¦ β π } ++ { β π π₯ , π₯ β π³ β π } + { β π π¦ , π¦ β π΄ β π } (3.42)is the ( π + π ) -phenomenon of the set of marginalprobabilities Λ π ( π |π³ + π΄ // π³ + π΄ ) = { π π₯ , π₯ β π³ } + { π π¦ , π¦ β π΄} (3.43) HE XIV FAMEMSβ2015 C
ONFERENCE for the sum π³ + π΄ .From previous formulas (3.32), (3.33), and (3.29) fora inserted pseudo-Kopula and conditional Kopulawe obtain formulas K (οΈ Λ π ( π | π + π// π³ + π΄ ) )οΈ == K ( π ) (οΈ Λ π ( π | π ( β© π// π΄ ) // π³ ( β© π// π΄ ) ) )οΈ , (3.44) K (οΈ Λ π ( π | π + π// π³ + π΄ ) )οΈ == K | π (οΈ Λ π ( π | π//
π³ |
π// π΄ ) )οΈ K (οΈ Λ π ( π | π// π΄ ) )οΈ , (3.45)that for each π β π΄ connect the Kopula of sum π³ + π΄ with the product of π -conditional Kopula π³ with respect to π΄ and the value of Kopula π΄ at π -phenomenon; and also with the π -inserted pseudo-Kopula of π³ which is inserted in the terraced event ter ( π // π΄ ) , generated by π΄ . The basis of the frame method of constructing Kop-ula is a rather simple idea of composing an arbi-trary π -s.e. X using the recurrence frame formula: X = { π₯ , π₯ , ..., π₯ π β } = { π₯ } + π³ , (3.46)where ( π β -s.e.βs π³ = X β { π₯ } = { π₯ , ..., π₯ π β } = (οΈ π³ β² (+) π³ β²β² )οΈ (3.47)are composed from two ( π β -s.e.βs π³ β² and π³ β²β² byset-theoretic operation of π -union and deο¬ned asthe inserted s.e.βs in the frame monoplet { π₯ } by thefollowing formulas: π³ β² = π³ ( β©{ π₯ } // { π₯ } ) == { π₯ } ( β© ) π³ = { π₯ β© π₯ , ..., π₯ β© π₯ π β } , π³ β²β² = π³ ( β©β // { π₯ } ) == { π₯ π } ( β© ) π³ = { π₯ π β© π₯ , ..., π₯ π β© π₯ π β } . (3.48)This simple idea allows us to ο¬nd the recurrentframe formulas for the π -Kopula of s.e. X asfunctions of the set of marginal probabilities Λ π = { π , π , . . . , π π β } .The frame method relies on formulas (3.16) and(3.17) and also correspondingly on (3.44) and (3.45),and constructs two recurrent formulas: K X (Λ π ) = Recursion (οΈ K ( { π₯ } ) π³ β² (Λ π ) , K ( β ) π³ β²β² (Λ π ) )οΈ , (3.49) K X (Λ π ) = Recursion (οΈ K |{ π₯ }π³ β² (Λ π ) , K |β π³ β²β² (Λ π ) , π )οΈ , (3.50) -intersection and -union are an intersection and union odsets by Minkowski (see details in [1]). for the π -Kopula of π -s.e. X through known proba-bility π of the event π₯ and together with it throughtwo known inserted pseudo- ( π β -Kopulas (see Def-inition 9), i.e., through pseudo- ( π β -Kopulas of in-serted ( π β -s.e.βs π³ β² and π³ β²β² in the frame mono-plet { π₯ } , either through two known conditional ( π β -Kopulas (see Deο¬nition 10) with respect tothe frame monoplet { π₯ } of the same π³ β² and π³ β²β² .Note, that for the sake of brevity in the formulas(3.49) and (3.50) we use the following abbrevia-tions, of course, given that X = π³ + { π₯ } : K X (Λ π ) = K (οΈ Λ π ( π | π + π// π³ + { π₯ } ) )οΈ , K ( { π₯ } ) π³ β² (Λ π ) = K ( { π₯ } ) (οΈ Λ π ( π | π ( β©{ π₯ } // { π₯ } ) // π³ β² ) )οΈ , K ( β ) π³ β²β² (Λ π ) = K ( β ) (οΈ Λ π ( π | π ( β©β // { π₯ } ) // π³ β²β² ) )οΈ , K |{ π₯ }π³ β² (Λ π ) = K |{ π₯ } (οΈ Λ π ( π | π//
π³ |{ π₯ } // { π₯ } ) )οΈ , K |β π³ β²β² (Λ π ) = K |β (οΈ Λ π ( π | π//
π³ |β // { π₯ } ) )οΈ . (3.51) Note 14 (about term ββframeββ).
Although in formu-las (3.46) and (3.47) only monoplet { π₯ } is a frame set, we shall call frame (with respect to this mono-plet) the s.e. X itself, construcyed from two inserteds.e.βs π³ β² = π³ ( β©{ π₯ } // { π₯ } ) and π³ β²β² = π³ ( β©β // { π₯ } ) , morehoping to clarify understanding than to cause mis-understandings. Note 15 (recurrent formulas of the frame method).
Getting rid of abbreviations (3.51) and using (3.44)and (3.45), we write the recurrent formulas of the frame method (3.49) and (3.50) in the expandedform: K (οΈ Λ π ( π | π + π// π³ + { π₯ } ) )οΈ == β§βͺβ¨βͺβ© K ( { π₯ } ) (οΈ Λ π ( π | π ( β©{ π₯ } // { π₯ } ) // π³ β² ) )οΈ , π = { π₯ } , K ( β ) (οΈ Λ π ( π | π ( β©β // { π₯ } ) // π³ β²β² ) )οΈ , π = β , (3.52) K (οΈ Λ π ( π | π + π// π³ + { π₯ } ) )οΈ == β§βͺβ¨βͺβ© K |{ π₯ } (οΈ Λ π ( π | π//
π³ |{ π₯ } // { π₯ } ) )οΈ π , π = { π₯ } , K |β (οΈ Λ π ( π | π//
π³ |β // { π₯ } ) )οΈ (1 β π ) , π = β . (3.53)Although formulas (3.46) and (3.47) are satisο¬ed forany s.e., but in the proposed frame method (3.49)and (3.50) we use only half-rare s.e. (s.h-r.e.) [6].This, however, does not detract from the generalityof its application, since the set-phenomenon trans-formations are any s.e. can be obtained from itshalf-rare projection [6].We will make the following useful Note 16 (any half-rare s.e. is composed by the framemethod from two inserted s.e.βs which are always half-rare).
If the frame s.e. X in (3.46) ans (3.47) is OROBYEV half-rare, i.e., its marginal probabilities from Λ π = { π , π , ..., π π β } are not more than half, forexample: / (cid:62) π (cid:62) π (cid:62) ... (cid:62) π π β , (3.54)then the both inserted s.e.βs π³ β² = { π₯ β² , ..., π₯ β² π β } and π³ β²β² = { π₯ β²β² , ..., π₯ β²β² π β } , and together with them and thes.e. π³ are also half-rare by its Deο¬nition (3.48). Inother words, their marginal probabilities from Λ π β² = { π β² , ..., π β² π β } Λ π β²β² = { π β²β² , ..., π β²β² π β } do not exceed thecorresponding marginal probabilities events fromthe frame s.e. X : π (cid:62) π β² , ..., π π β (cid:62) π β² π β ,π (cid:62) π β²β² , ..., π π β (cid:62) π β²β² π β , (3.55)and marginal probabilities from Λ π π β = { π , ..., π π β } are half-rare by deο¬nition. Thus,any half-rare π -s.e. is composed by the framemethod with the formula (3.47) from two inserted ( π β -s.e.βs π³ β² and π³ β²β² , which are required to behalf-rare. Lemma 4 (about independent half-rare s.e.βs , con-structed by the frame method from two inserted half-rares.e.βs ). That in the family of half-rare s.e.βs X withsets of marginal probabilitiues Λ π , constructed bythe frame method from two inserted half-rare s.e.βs π³ β² and π³ β²β² , there was an independent half-rare s.e. ,it is necessary so that the sets of marginal probabil-ities are related to the marginal probabilities of theframe s.e. X = { π₯ } + π³ = { π₯ } + ( π³ β² (+) π³ β²β² ) by thefollowing way: Λ π β² = {οΈ π β² , . . . , π β² π β }οΈ = { π π , . . . , π π β π } , Λ π β²β² = {οΈ π β²β² , . . . , π β²β² π β }οΈ = { π (1 β π ) , . . . , π π β (1 β π ) } ; (3.56)and suο¬cient so that the e.p.d. of the 1st kind ofinserted s.e.βs π³ β² and π³ β²β² to be calculated from theformulas for π β π³ : π ( π β² // π³ β² ) = P (οΈ βοΈ π₯ β² β π β² π₯ β² βοΈ π₯ β² βπ³ β² β π β² π₯ β² π )οΈ == β§βͺβͺβ¨βͺβͺβ© π βοΈ π₯ β π π π₯ βοΈ π₯ βπ³ β π (1 β π π₯ ) , π ΜΈ = β ,π βοΈ π₯ βπ³ (1 β π π₯ ) + 1 β π , π = β ,π ( π β²β² // π³ β²β² ) = P (οΈ βοΈ π₯ β²β² β π β²β² π₯ β²β² βοΈ π₯ β²β² βπ³ β²β² β π β²β² π₯ β²β² π )οΈ == β§βͺβͺβ¨βͺβͺβ© (1 β π ) βοΈ π₯ β π π π₯ βοΈ π₯ βπ³ β π (1 β π π₯ ) , π ΜΈ = β , (1 β π ) βοΈ π₯ βπ³ (1 β π π₯ ) + π , π = β , (3.57)where π β² = { π₯ β² , π₯ β π } = { π₯ β© π₯, π₯ β π } β π³ β² ,π β²β² = { π₯ β²β² , π₯ β π } = { π₯ π β© π₯, π₯ β π } β π³ β²β² . (3.58) Proof . The necessity is obvious, since the insertedmarginal probabilities of the independent s.e. X are probabilities of double intersections of inde-pendent events which have the required form for π = 1 , ..., π β : π β² π = P ( π₯ β© π₯ π ) = π π π ,π β²β² π = P ( π₯ π β© π₯ π ) = π π (1 β π ) . (3.59)The sο¬ciency follows from (3.57) and formulasthat connect the e.p.d. of the 1st kind of frame s.e. X with the e.p.d. of the 1st kind of inserted s.e.βs π³ β² and π³ β²β² , which have the form for π β π³ : π ( π + π // π³ + { π₯ } ) == β§βͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβ© π ( π β² // π³ β² ) , π = { π₯ } , π ΜΈ = β ,π ( β // π³ β² ) β π , π = { π₯ } , π = β ,π ( π β²β² // π³ β²β² ) , π = β , π ΜΈ = β ,π ( β // π³ β²β² ) β π , π = β , π = β . (3.60)Demanding (3.60) to perform suο¬cient conditions(3.57), we get π ( π + π // π³ + { π₯ } ) == β§βͺβͺβͺβͺβ¨βͺβͺβͺβͺβ© π βοΈ π₯ β π π π₯ βοΈ π₯ βπ³β π (1 β π π₯ ) , π = { π₯ } , π β π³ , (1 β π ) βοΈ π₯ β π π π₯ βοΈ π₯ βπ³β π (1 β π π₯ ) ,π = β , π β π³ . (3.61)As a result, for the s.e. X we have the e.p.d. of the1st kind of independent events: π ( π + π // π³ + { π₯ } ) == π ( π// X ) = βοΈ π₯ β π π π₯ βοΈ π₯ β X β π (1 β π π₯ ) , (3.62)where π = {οΈ π + { π₯ } , π = { π₯ } , π β π³ ,π, π = β , π β π³ . (3.63)The lemma is proved. Theory of the Kopula of monoplets of events (1-Kopula) seemed to be completed by the formula(2.35). This formula deο¬nes the 1-Kopula of an ar-bitrary monoplet of events { π₯ } with { π₯ } -monopletof marginal probabilities Λ π = { π π₯ } β [0 , π₯ in theunique form: K (Λ π ) = K ( π π₯ ) = π π₯ , (4.1) By the way, the necessary condition also follows from (3.57). HE XIV FAMEMSβ2015 C
ONFERENCE which which provides 2 values on each ( π | Λ π ) -phenomenon-dom by general formulas for π β{ π₯ } : K (οΈ Λ π ( π | π// { π₯ } ) )οΈ == {οΈ K (1 β π π₯ ) = 1 β π π₯ , π = β , K ( π π₯ ) = π π₯ , π = { π₯ } . (4.2)However, the formula (4.2) can be generalized inthe following simple way: K (οΈ Λ π ( π | π// { π₯ } ) )οΈ == {οΈ K β (1 β π π₯ ) = 1 β K { π₯ } ( π π₯ ) , π = β , K { π₯ } ( π π₯ ) , π = { π₯ } , (4.3)where K { π₯ } is any function such that in half-rarevariables: K { π₯ } : [0 , / β [0 , / , (4.4)and in free variables: K { π₯ } : [0 , β [0 , . (4.5)In this case, the 1-Kopula (4.2) is an important spe-cial case of 1-Kopula (4.3) when K { π₯ } ( π π₯ ) = π π₯ . Thiscase corresponds to a uniform marginal distribu-tion function on the unit interval in the theory ofthe classical copula [2]. In order to construct by the frame method the Λ π - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered frame half-rare doublet of events X = { π₯, π¦ } = { π₯ } + π³ = { π₯ } + (οΈ π³ β² (+) π³ β²β² )οΈ (5.1)with the X -set of marginal probabilities Λ π = { π π₯ , π π¦ } ,where / (cid:62) π π₯ (cid:62) π π¦ (cid:62) , (5.2)letβs suppose that we have at our disposal two half-rare inserted monoplets of events π³ β² = { π₯ β© π¦ } = { π β² } π³ β²β² = { π₯ π β© π¦ } = { π β²β² } , (5.3)with known 1-Kopulas: K β² (οΈ Λ π ( π | π// { π β² } ) )οΈ == {οΈ K β² (1 β π π β² ) = 1 β π π β² , π = β , K β² ( π π β² ) = π π β² , π = { π β² } , (5.4) K β²β² (οΈ Λ π ( π | π// { π β²β² } ) )οΈ == {οΈ K β²β² (1 β π π β²β² ) = 1 β π π β²β² , π = β , K β²β² ( π π β²β² ) = π π β²β² , π = { π β²β² } , (5.5)By Deο¬nition of inserted monoplets (5.3) (see Fig.7) π β² = π₯ β© π¦ β π₯,π β²β² = π₯ π β© π¦ β π₯ π , (5.6)and also because of the Λ π -ordering assumption(5.2), we get that π π β² + π π β²β² = π π¦ (cid:54) π π₯ (cid:54) / (cid:54) β π π₯ . (5.7)Consequently, the 1-Kopulas of inserted mono-planes (5.4) and (5.5) are bound by the sum of theirmarginal probabilities:Consequently, the 1-Kopulas of inserted monoplets(5.4) and (5.5) are bound by the restriction on thesum of their marginal probabilities: π π β² + π π β²β² = π π¦ , (5.8)and depend on only one parameter: π π β² β [0 , π π¦ ] . (5.9) Ξ© π₯π¦ π β² π β© π₯ = π¦ π β© π₯ π β² = π₯ β© π¦ π β²β² = π₯ π β© π¦π β²β² π β© π₯ π = π¦ π β© π₯ π Ξ© π β² π β© π₯ = π¦ π β© π₯ π β² = π₯ β© π¦ π β²β² = π₯ π β© π¦π β²β² π β© π₯ π = π¦ π β© π₯ π β β π₯ β β π₯ π Figure 7: Venn diagrams of the frame half-rare doublet of events X = { π₯, π¦ } , / (cid:62) π π₯ (cid:62) π π¦ (up), and two inserted monoplets π³ β² = { π β² } and π³ β²β² = { π β²β² } (down) agreed with the frame doublet X in the followingsense: π¦ = π β² + π β²β² π β² β π₯, π β²β² β π₯ π . OROBYEV We get the following formulas: π ( π₯π¦// { π₯, π¦ } ) = π ( π β² // π³ β² ) = π π β² ,π ( π₯// { π₯, π¦ } ) = π ( β // π³ β² ) β π π₯ = π π₯ β π π β² ,π ( π¦// { π₯, π¦ } ) = π ( π β²β² // π³ β²β² ) = π π¦ β π π β² ,π ( β // { π₯, π¦ } ) = π ( β // π³ β²β² ) β π π₯ = 1 β π π¦ β π π₯ + π π β² . (5.10)These formulas express the e.p.d. of the 1st kindof the Λ π -ordered half-rare frame doublet of events X = { π₯, π¦ } through the e.p.d. of the 1st kindof inserted monoplets π³ β² and π³ β²β² , and the prob-ability of frame event π₯ , and, in the ο¬nal result,through their marginal probabilities π π₯ and π π¦ , andmarginal probability π π β² of the inserted monoplet π³ β² = { π β² } = { π₯ β© π¦ } .The formulas (5.10) express values of the 2-Kopulaof Λ π -ordered doublet X = { π₯, π¦ } through 1-Kopulasof inserted monoplets π³ β² = { π β² } and π³ β²β² = { π β²β² } .Rewrite this in a form of an explicit recurrent for-mula: π ( π// { π₯, π¦ } ) = K X (οΈ Λ π ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβͺβ¨βͺβͺβͺβͺβ© K π³ β² ( π π β² ) , π = { π₯, π¦ } , K π³ β² (1 β π π β² ) β π π₯ , π = { π₯ } , K π³ β²β² ( π π β²β² ) , π = { π¦ } , K π³ β²β² (1 β π π β²β² ) β π π₯ , π = β . (5.11)Considering (2.34) and (5.9), we will continue: π ( π// { π₯, π¦ } ) = K X (οΈ Λ π ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβͺβ¨βͺβͺβͺβͺβ© π π β² , π = { π₯, π¦ } ,π π₯ β π π β² , π = { π₯ } ,π π β²β² , π = { π¦ } , β π π₯ β π π β²β² , π = β , (5.12) = β§βͺβͺβͺβͺβ¨βͺβͺβͺβͺβ© π π β² , π = { π₯, π¦ } ,π π₯ β π π β² , π = { π₯ } ,π π¦ β π π β² , π = { π¦ } , β π π₯ β π π¦ + π π β² , π = β . (5.13)We note, by the way, that the restriction (5.9) bythe assumption of Λ π -ordered (5.2) is a special caseof FrΒ΄echet-inequalities: (cid:54) π π β² (cid:54) π + π₯π¦ = min { π π₯ , π π¦ } = π π¦ . (5.14) Note 17 (frame mathod for otherwise Λ π -ordered half-rare doublet of events). For otherwise Λ π -ordered half-rare doublet of events X = { π¦, π₯ } = { π¦ } + π³ = { π¦ } + (οΈ π³ β² (+) π³ β²β² )οΈ , (5.15)where π³ β² = { π¦ β© π₯ } = { π β² } π³ β²β² = { π¦ π β© π₯ } = { π β²β² } , (5.16) Ξ© π¦π₯ π β² π β© π¦ = π₯ π β© π¦ π β² = π¦ β© π₯ π β²β² = π¦ π β© π₯π β²β² π β© π¦ π = π₯ π β© π¦ π Ξ© π β² π β© π¦ = π₯ π β© π¦ π β² = π¦ β© π₯ π β²β² = π¦ π β© π₯π β²β² π β© π¦ π = π₯ π β© π¦ π β β π¦ β β π¦ π Figure 8: Venn diagrams of the frame, otherwise ordered half-rare dou-blet events X = { π¦, π₯ } , / (cid:62) π π¦ (cid:62) π π₯ (up), and two inserted monoplets π³ β² = { π β² } and π³ β²β² = { π β²β² } (down), agreed with the frame doublet X inthe following sense: π₯ = π β² + π β²β² π β² β π¦, π β²β² β π¦ π . with the X -set of marginal probabilities Λ π = { π π¦ , π π₯ } , / (cid:62) π π¦ (cid:62) π π₯ , the assumptions (5.18) take symmet-rical form: π β² = π¦ β© π₯ β π¦,π β²β² = π¦ π β© π₯ β π¦ π . (5.17)By Deο¬nition of inserted monoplets (5.16) (. . 8) π β² = π₯ β© π¦ β π¦,π β²β² = π₯ β© π¦ π β π¦ π , (5.18)and also because of the assumption and also be-cause of the assumption of another Λ π -ordering, weget that π π β² + π π β²β² = π π₯ (cid:54) π π¦ (cid:54) / (cid:54) β π π¦ . (5.19)Consequently, 1-copulas of inserted monoplanesare connected by a restriction on the sum of theirmarginal probabilities: π π β² + π π β²β² = π π₯ , (5.20)and depend on only one parameter: π π β² β [0 , π π₯ ] . (5.21)By the assumptions, the following formulas are HE XIV FAMEMSβ2015 C
ONFERENCE valid: π ( π₯π¦// { π¦, π₯ } ) = π ( π β² // π³ β² ) = π π β² ,π ( π¦// { π¦, π₯ } ) = π ( β // π³ β² ) β π π¦ = π π¦ β π π β² ,π ( π₯// { π¦, π₯ } ) = π ( π β²β² // π³ β²β² ) = π π₯ β π π β² ,π ( β // { π¦, π₯ } ) = π ( β // π³ β²β² ) β π π¦ = 1 β π π₯ β π π¦ β π π β² . (5.22)These formulas express the e.p.d. of the 1stkind otherwise Λ π -ordered half-rare frame doubletof events X through the e.p.d. of the 1st kind of inserted monoplets π³ β² and π³ β²β² , and the probabilityof frame event π¦ , and, in the ο¬nal result, throughown marginal probabilities π π₯ π π¦ , and the marginalprobabilities of inserted monoplet π³ β² = { π β² } = { π₯ β© π¦ } (see Fig. 8).The formulas (5.22) express values of the 2-Kopulaof Λ π -ordered doublet X = { π¦, π₯ } through 1-Kopulasof inserted monoplets π³ β² = { π β² } and π³ β²β² = { π β²β² } .Rewrite this in the form of explicit recurrent for-mula: π ( π// { π¦, π₯ } ) = K X (οΈ Λ π ( π | π// { π¦,π₯ } ) )οΈ == β§βͺβͺβͺβ¨βͺβͺβͺβ© K π³ β² ( π π β² ) , π = { π₯, π¦ } , K π³ β² (1 β π π β² ) β π π¦ , π = { π¦ } , K π³ β²β² ( π π β²β² ) , π = { π₯ } , K π³ β²β² (1 β π π β²β² ) β π π¦ , π = β . (5.23)Continue: π ( π// { π₯, π¦ } ) = K X (οΈ Λ π ( π | π// { π₯,π¦ } ) )οΈ == β§βͺβͺβͺβ¨βͺβͺβͺβ© π π β² , π = { π₯, π¦ } ,π π¦ β π π β² , π = { π¦ } ,π π β²β² , π = { π₯ } , β π π₯ β π π β²β² , π = β , (5.24) = β§βͺβͺβͺβ¨βͺβͺβͺβ© π π β² , π = { π₯, π¦ } ,π π¦ β π π β² , π = { π¦ } ,π π₯ β π π β² , π = { π₯ } , β π π₯ β π π¦ + π π β² , π = β . (5.25)We note, as above, that the restriction (5.21) by theassumption of another Λ π -ordering is a special caseof FrΒ΄echet inequalities: (cid:54) π π β² (cid:54) π + π₯π¦ = min { π π₯ , π π¦ } = π π₯ . (5.26) The formulas (5.11), as well as formulas (5.13), canbe rewrite in the form of special cases of recurrentformulas (3.52) and (3.53) from Note 15 for the dou-blet X = { π₯, π¦ } = { π₯ } + { π¦ } = { π₯ } + π³ : K (οΈ Λ π ( π | π + π// π³ + { π₯ } ) )οΈ == β§βͺβ¨βͺβ© K ( { π₯ } ) (οΈ Λ π ( π | π ( β©{ π₯ } // { π₯ } ) // π³ β² ) )οΈ , π = { π₯ } , K ( β ) (οΈ Λ π ( π | π ( β©β // { π₯ } ) // π³ β²β² ) )οΈ , π = β , (5.27) K (οΈ Λ π ( π | π + π// π³ + { π₯ } ) )οΈ == β§βͺβ¨βͺβ© K |{ π₯ } (οΈ Λ π ( π | π//
π³ |{ π₯ } // { π₯ } ) )οΈ π π₯ , π = { π₯ } , K |β (οΈ Λ π ( π | π//
π³ |β // { π₯ } ) )οΈ (1 β π π₯ ) , π = β . (5.28)In the formulas (5.27) the pseudo-Kopulas K ( { π₯ } ) and K ( β ) of inserted monoplets π³ β² and π³ β²β² , corre-spondingly, are deο¬ned by the ο¬rst and the secondpairs of probabilities from (5.13) correspondingly,i.e., by formulas: K ( { π₯ } ) (οΈ Λ π ( π | π ( β©{ π₯ } // { π₯ } ) // π³ β² ) )οΈ == {οΈ π π β² , π = { π₯, π¦ } ,π π₯ β π π β² , π = { π₯ } , (5.29) K ( β ) (οΈ Λ π ( π | π ( β©β // { π₯ } ) // π³ β²β² ) )οΈ == {οΈ π π¦ β π π β² , π = { π¦ } , β π π₯ β π π¦ + π π β² , π = β , (5.30)where, for example, for π = { π¦ }{ π¦ } ( β©{ π₯ } // { π₯ } ) = { π¦ β© π₯ } β π³ β² = { π β² } , { π¦ } ( β©β // { π₯ } ) = { π¦ β© π₯ π } β π³ β²β² = { π¦ β π β² } , (5.31)and the corresponding sets of marginal probabili-ties of inserted monoplets π³ β² and π³ β²β² have the form Λ π ( π |{ π¦ } ( β©{ π₯ } // { π₯ } ) // π³ β² ) = { π π β² } , Λ π ( π |{ π¦ } ( β©β // { π₯ } ) // π³ β²β² ) = { π π¦ β π π β² } . (5.32)In the formulas (5.28) the conditional Kopulas K |{ π₯ } and K |β are deο¬ned by the ο¬rst and the secondpairs of probabilities from (5.13), normalized by π π₯ and by β π π₯ correspondingly, i.e. by the formulas: K |{ π₯ } (οΈ Λ π ( π | π//
π³ |{ π₯ } // { π₯ } ) )οΈ == {οΈ π π₯ π π β² , π = { π₯, π¦ } , π π₯ ( π π₯ β π π β² ) , π = { π₯ } , (5.33) K |β (οΈ Λ π ( π | π//
π³ |β // { π₯ } ) )οΈ == {οΈ β π π₯ ( π π¦ β π π β² ) , π = { π¦ } , β π π₯ (1 β π π₯ β π π¦ + π π β² ) , π = β . (5.34)The corresponding sets of marginal conditionalprobabilities of events π¦ β π³ with respect tothe frame terraced events ter ( { π₯ } // { π₯ } ) = π₯ ter ( β // { π₯ } ) = π₯ π correspondingly have the form: Λ π ( π |π³ // π³ |{ π₯ } // { π₯ } ) = {οΈ π π β² π π₯ }οΈ , Λ π ( π |π³ // π³ |β // { π₯ } ) = {οΈ π π¦ β π π β² β π π₯ }οΈ . (5.35) OROBYEV Remind, that FrΒ΄echet restrictions on the functionalparameter π π β² = π π β² ( π π₯ , π π¦ ) for Λ π -ordered half-raredoublet of events X = { π₯, π¦ } have the form: (cid:54) π π β² (cid:54) π + π₯π¦ = min { π π₯ , π π¦ } = π π¦ , (5.36)and for otherwise Λ π -ordered half-rare doublet ofevents X = { π¦, π₯ } have the form: (cid:54) π π β² (cid:54) π + π₯π¦ = min { π π₯ , π π¦ } = π π₯ . (5.37) -Kopula First, without the frame method, which is not re-quired here, consider the simplest example of a -Kopula K β Ξ¨ X of the ( π β -set of events X = { π₯, π¦, π§ } , i.e., a 1-function on the unit X -cube. In other words, construct such a nonnegativebounded numerical function K : [0 , β X β [0 , , that for all π§ β X βοΈ π₯ β π β X K (οΈ Λ π€ ( π | π// X ) )οΈ = π€ π₯ . Such a simple example of a 1-function on X -cube isso-called independent ( π β -Kopula , which for allfree variables Λ π€ = { π€ π₯ , π€ π¦ , π€ π§ } β [0 , π₯ β [0 , π¦ β [0 , π§ = [0 , β X is deο¬ned by the formula: K ( Λ π€ ) = π€ π₯ π€ π¦ π€ π§ , (6.1)that provides it on each ( π | Λ π€ ) -phenomenon-domthe following values: K (οΈ Λ π€ ( π | π// X ) )οΈ = βοΈ π₯ β π π€ π₯ βοΈ π₯ β X β π (1 β π€ π₯ ) (6.2)for π β X . Indeed, as in the case of the doubletof events this function is a 1-function, since for all π₯ β X βοΈ π₯ β π β X (οΈ βοΈ π§ β π π€ π§ βοΈ π§ β X β π (1 β π€ π§ ) )οΈ = π€ π₯ . The e.p.d. of the 1st kind of independent triplet ofevents X with the X -set of probabilities of events Λ π is deο¬ned by values of the independent 3-Kopula(6.1) on ( π | Λ π ) -phenomenon-dom by the general for-mulas of half-rare variables, i.e., for π β { π₯, π¦ } : π ( π// X ) = K (οΈ Λ π ( π | π// X ) )οΈ = βοΈ π₯ β π π π₯ βοΈ π₯ β X β π (1 β π π₯ ) . (6.3) In Fig. 9 it is shown the results of visualizationof the three-dimensional graph of independent 3-Kopula (8.1) of the triplet X = { π₯, π¦, π§ } , deο¬ned onthe cube [0 , , in projections on planes, which areorthogonal to the axis π π¦ . Figure 9: The visualization of projections of the same three-dimensionalmap of Cartesian representation of independent 3-Kopula of the triplet X = { π₯, π¦, π§ } on the unit cube in conditional colors with values ofmarginal probaboility π π¦ = 0 . , ..., . , . , where the white color cor-responds to points in which probabilities of all terraced events are 1/8.The orientation of axes: ( π π₯ , π π§ ) = (horizontal, vertical). In order to construct by the frame method the Λ π - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered frame half-rare triplet of events X = { π₯, π¦, π§ } with the X -set of marginal probabilities Λ π = { π π₯ , π π¦ , π π§ } , where / (cid:62) π π₯ (cid:62) π π¦ (cid:62) π π§ (cid:62) , (6.4)letβs suppose that X = { π₯ } + { π¦, π§ } = { π₯ } + ( π³ β² (+) π³ β²β² ) (6.5)and in our disposal we have two inserted half-raredoublets of events π³ β² = { π β² , π‘ β² } π³ β²β² = { π β²β² , π‘ β²β² } , with the known 2-Kopulas (see Fig. 10) which bythe deο¬nition satisfy the following inclusions: π β² = π₯ β© π¦ β π₯, π‘ β² = π₯ β© π§ β π₯, π β² βͺ π‘ β² β π₯,π β²β² = π₯ π β© π¦ β π₯ π , π‘ β²β² = π₯ π β© π§ β π₯ π , π β²β² βͺ π‘ β²β² β π₯ π . (6.6)In view of the assumptions made (see Fig. 10) ter ( π₯π¦π§// X ) = π₯ β© π¦ β© π§ = π β² β© π‘ β² = ter ( π β² π‘ β² // π³ β² ) , ter ( π₯π¦// X ) = π₯ β© π¦ β© π§ π = π β² β© π‘ β² π = ter ( π β² // π³ β² ) , ter ( π₯π§// X ) = π₯ β© π¦ π β© π§ = π β² π β© π‘ β² = ter ( π‘ β² // π³ β² ) , ter ( π¦π§// X ) = π₯ π β© π¦ β© π§ = π β²β² β© π‘ β²β² = ter ( π β²β² π‘ β²β² // π³ β²β² ) , ter ( π¦// X ) = π₯ π β© π¦ β© π§ π = π β²β² β© π‘ β²β² π = ter ( π β²β² // π³ β²β² ) , ter ( π§// X ) = π₯ π β© π¦ π β© π§ = π β²β² π β© π‘ β²β² = ter ( π‘ β²β² // π³ β²β² ) , (6.7)these 6 terraced events are deο¬ned. All of them aregenerated by the frame half-rare triplet X , with the
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ONFERENCE Ξ© π₯ π β© π β²β² π β© π‘ β²β² π π¦π₯ π§ π β²β² β© π‘ β²β² π π β² β© π‘ β² π π₯ β© π β² π β© π‘ β² π π β² β© π‘ β² π β²β² β© π‘ β²β² π β² π β© π‘ β² π β²β² π β© π‘ β²β² Ξ© π₯ β© π β² π β© π‘ β² π π₯ π β© π β²β² π β© π‘ β²β² π π β² β© π‘ β² π π β²β² β© π‘ β²β² π π β² β© π‘ β² π β²β² β© π‘ β²β² π β² π β© π‘ β² π β²β² π β© π‘ β²β² π β² π β²β² π‘ β² π‘ β²β² β β π₯ β β π₯ π Figure 10: Venn diagrams of the frame half-rare trip0let of events X = { π₯, π¦, π§ } , / (cid:62) π π₯ (cid:62) π π¦ (cid:62) π π§ (up), and two inserted doublets π³ β² = { π β² , π‘ β² } and π³ β²β² = { π β²β² , π‘ β²β² } (down), agreed with the frame triplet X in thefollowing sense: π¦ = π β² + π β²β² , π§ = π‘ β² + π‘ β²β² π β² βͺ π‘ β² β π₯, π β²β² βͺ π‘ β²β² β π₯ π . exception of two terraced events ter ( π₯// X ) = π₯ β© π¦ π β© π§ π , ter ( β // X ) = π₯ π β© π¦ π β© π§ π , (6.8)that are deο¬ned by the formulas: ter ( π₯// X ) = π₯ β π β² βͺ π‘ β² , ter ( β // X ) = π₯ π β π β²β² βͺ π‘ β²β² , (6.9)or, equivalently, ter ( π₯// X ) = ter ( β // π³ β² ) β π₯ π , ter ( β // X ) = ter ( β // π³ β²β² ) β π₯. (6.10)In view of this, we obtain the formulas: π ( π₯π¦π§// X ) = π ( π β² π‘ β² // π³ β² ) ,π ( π₯π¦// X ) = π ( π β² // π³ β² ) ,π ( π₯π§// X ) = π ( π‘ β² // π³ β² ) ,π ( π₯// X ) = π ( β // π³ β² ) β π π₯ ,π ( π¦π§// X ) = π ( π β²β² π‘ β²β² // π³ β²β² ) ,π ( π¦// X ) = π ( π β²β² // π³ β²β² ) ,π ( π§// X ) = π ( π‘ β²β² // π³ β²β² ) ,π ( β // X ) = π ( β // π³ β²β² ) β π π₯ , (6.11) that express the e.p.d. of the 1st kind of frame half-rare triplet of events X through the e.p.d. of the 1stkind of inserted half-rare doublets π³ β² and π³ β²β² , andthe probability of frame event π₯ .In the language of e.p.d. of the 1st kind assump-tions (6.6) mean that π ( π β² // π³ β² ) + π ( π β² π‘ β² // π³ β² )++ π ( π β²β² // π³ β² ) + π ( π β²β² π‘ β²β² // π³ β² ) = π π¦ ,π ( π‘ β² // π³ β² ) + π ( π β² π‘ β² // π³ β² )++ π ( π‘ β²β² // π³ β² ) + π ( π β²β² π‘ β²β² // π³ β² ) = π π§ , (6.12)or the same in the language of probabilities events: π π β² + π π β²β² = π π¦ ,π π‘ β² + π π‘ β²β² = π π§ , (6.13)In addition, the third pair of inclusions under theassumptions (6.6) means that π π β² + π π‘ β² β π π β² π‘ β² (cid:54) π π₯ ,π π β²β² + π π‘ β²β² β π π β²β² π‘ β²β² (cid:54) β π π₯ , (6.14)where π π β² π‘ β² = π ( π β² π‘ β² // π³ β² ) = P ( π β² β© π‘ β² ) ,π π β²β² π‘ β²β² = π ( π β²β² π‘ β²β² // π³ β²β² ) = P ( π β²β² β© π‘ β²β² ) (6.15)are probabilities of double intersections of eventsfrom the inserted doublets π³ β² and π³ β²β² .Taking into account the FrΒ΄echet inequalities the re-strictions (6.13) and (6.14) are equivalent to thefollowing inequalities for 4 parameters π π β² , π π‘ β² , π π β² π‘ β² π π β²β² π‘ β²β² of inserted doublets π³ β² and π³ β²β² : (cid:54) π π β² (cid:54) π π¦ , (cid:54) π π‘ β² (cid:54) π π§ ,π β π β² π‘ β² (cid:54) π π β² π‘ β² (cid:54) π + π β² π‘ β² ,π β π β²β² π‘ β²β² (cid:54) π π β²β² π‘ β²β² (cid:54) π + π β²β² π‘ β²β² , (6.16)where π β π β² π‘ β² = max { , π π β² + π π‘ β² β π π₯ } ,π + π β² π‘ β² = min { π π β² , π π‘ β² } ,π β π β²β² π‘ β²β² = max { , π π₯ + π π¦ + π π§ β β π π β² β π π‘ β² } ,π + π β²β² π‘ β²β² = min { π π¦ β π π β² , π π§ β π π‘ β² } , (6.17)are the lower and upper Frechet-boundaries ofprobabilities of double intersections of inserteddoublets π³ β² and π³ β²β² with respect to the frame mono-plet { π₯ } .Letβs write the formulas (6.11), using only these4 parameters and remembering the restrictions OROBYEV (6.16): π ( π₯π¦π§// X ) = π π β² π‘ β² ,π ( π₯π¦// X ) = π π β² β π π β² π‘ β² ,π ( π₯π§// X ) = π π‘ β² β π π β² π‘ β² ,π ( π₯// X ) = π π₯ β π π β² β π π‘ β² + π π β² π‘ β² ,π ( π¦π§// X ) = π π β²β² π‘ β²β² ,π ( π¦// X ) = π π¦ β π π β² β π π β²β² π‘ β²β² ,π ( π§// X ) = π π§ β π π‘ β² β π π β²β² π‘ β²β² ,π ( β // X ) = 1 β π π₯ β π π¦ β π π§ + π π β² + π π‘ β² + π π β²β² π‘ β²β² . (6.18) The formulas (6.18) as well as the formulas (6.11)can be written in the form of special cases of recur-rence formulas (3.52) and (3.53) from Note 15 forthe triplet X = { π₯, π¦, π§ } = { π₯ } + { π¦, π§ } = { π₯ } + π³ : K (οΈ Λ π ( π | π + π// π³ + { π₯ } ) )οΈ == β§βͺβ¨βͺβ© K ( { π₯ } ) (οΈ Λ π ( π | π ( β©{ π₯ } // { π₯ } ) // π³ β² ) )οΈ , π = { π₯ } , K ( β ) (οΈ Λ π ( π | π ( β©β // { π₯ } ) // π³ β²β² ) )οΈ , π = β , (6.19) K (οΈ Λ π ( π | π + π// π³ + { π₯ } ) )οΈ == β§βͺβ¨βͺβ© K |{ π₯ } (οΈ Λ π ( π | π//
π³ |{ π₯ } // { π₯ } ) )οΈ π π₯ , π = { π₯ } , K |β (οΈ Λ π ( π | π//
π³ |β // { π₯ } ) )οΈ (1 β π π₯ ) , π = β . (6.20)In the formulas (6.19) the inserted pseudo-Kopulas K ( { π₯ } ) and K ( β ) are deο¬ned by the ο¬rst and the sec-ond four probabilities from (6.18) correspondingly,i.e., by the formulas: K ( { π₯ } ) (οΈ Λ π ( π | π ( β©{ π₯ } // { π₯ } ) // π³ β² ) )οΈ == β§βͺβͺβͺβ¨βͺβͺβͺβ© π π β² π‘ β² , π = { π¦, π§ } ,π π β² β π π β² π‘ β² , π = { π¦ } ,π π‘ β² β π π β² π‘ β² , π = { π§ } ,π π₯ β π π β² β π π‘ β² + π π β² π‘ β² , π = β , (6.21) K ( β ) (οΈ Λ π ( π | π ( β©β // { π₯ } ) // π³ β²β² ) )οΈ == β§βͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβ© π π β²β² π‘ β²β² , π = { π¦, π§ } ,π π¦ β π π β² β π π β²β² π‘ β²β² , π = { π¦ } ,π π§ β π π‘ β² β π π β²β² π‘ β²β² , π = { π§ } , β π π₯ β π π¦ β π π§ ++ π π β² + π π‘ β² + π π β²β² π‘ β²β² , π = β , (6.22)where, for example, for π = { π¦, π§ }{ π¦, π§ } ( β©{ π₯ } // { π₯ } ) = { π¦ β© π₯, π§ β© π₯ } β π³ β² = { π β² , π‘ β² } , { π¦, π§ } ( β©β // { π₯ } ) = { π¦ β© π₯ π , π§ β© π₯ π } β π³ β²β² = { π β²β² , π‘ β²β² } , (6.23) and the corresponding sets of marginal probabili-ties of inserted doublets π³ β² π³ β²β² have the form Λ π ( π |{ π¦,π§ } ( β©{ π₯ } // { π₯ } ) // π³ β²β² ) = { π π β² , π π‘ β² } , Λ π ( π |{ π¦,π§ } ( β©β // { π₯ } ) // π³ β²β² ) = { π π¦ β π π β² , π π§ β π π‘ β² } . (6.24)In the formulas (6.20) the conditional Kopulas K |{ π₯ } and K |β are deο¬ned by the ο¬rst and the second fourprobabilities from (6.18), normalized by π π₯ and by β π π₯ correspondingly, i.e., by the formulas: K |{ π₯ } (οΈ Λ π ( π | π//
π³ |{ π₯ } // { π₯ } ) )οΈ == β§βͺβͺβͺβͺβ¨βͺβͺβͺβͺβ© π π₯ π π β² π‘ β² , π = { π¦, π§ } , π π₯ ( π π β² β π π β² π‘ β² ) , π = { π¦ } , π π₯ ( π π‘ β² β π π β² π‘ β² ) , π = { π§ } , β π π₯ ( π π β² + π π‘ β² β π π β² π‘ β² ) , π = β , (6.25) K |β (οΈ Λ π ( π | π//
π³ |β // { π₯ } ) )οΈ == β§βͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβ© β π π₯ π π β²β² π‘ β²β² , π = { π¦, π§ } , β π π₯ ( π π¦ β π π β² β π π β²β² π‘ β²β² ) , π = { π¦ } , β π π₯ ( π π§ β π π‘ β² β π π β²β² π‘ β²β² ) , π = { π§ } , β π π₯ (1 β π π₯ β π π¦ β π π§ )++ β π π₯ ( π π β² + π π‘ β² + π π β²β² π‘ β²β² ) , π = β . (6.26)The corresponding sets of marginal conditionalprobabilities of events π¦, π§ β π³ with respectto the frame terraced events ter ( { π₯ } // { π₯ } ) = π₯ ter ( β // { π₯ } ) = π₯ π correspondingly have the from: Λ π ( π |π³ // π³ |{ π₯ } // { π₯ } ) = {οΈ π π β² π π₯ , π π‘ β² π π₯ }οΈ , Λ π ( π |π³ // π³ |β // { π₯ } ) = {οΈ π π¦ β π π β² β π π₯ , π π§ β π π‘ β² β π π₯ }οΈ . (6.27)Remind, that the four functional parameters π π β² , π π‘ β² , π π β² π‘ β² and π π β²β² π‘ β²β² in the recurrent formulas(6.19), (6.20), and also in the formulas for pseudo-Kopulas (6.21), (6.22), and the conditional Kopulas(6.25), (6.26), obey the Frechet-constraints (6.16). In order by the frame method to construct the Λ π - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered frame half-rare quadruplet of events X = { π₯, π¦, π§, π£ } with the X -set of marginal probabilities Λ π = { π π₯ , π π¦ , π π§ , π π£ } , where / (cid:62) π π₯ (cid:62) π π¦ (cid:62) π π§ (cid:62) π π£ (cid:62) , (7.1)letβs suppose that X = { π₯ } + { π¦, π§, π£ } = { π₯ } + ( π³ β² (+) π³ β²β² ) (7.2)
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ONFERENCE and we have two inserted half-rare triplets of events π³ β² = { π β² , π‘ β² , π’ β² } π³ β²β² = { π β²β² , π‘ β²β² , π’ β²β² } , with the known 3-Kopulas, which by deο¬nition sat-isfy the following inclusions (see Fig. 11): π β² = π₯ β© π¦ β π₯, π‘ β² = π₯ β© π§ β π₯, π’ β² = π₯ β© π£ β π₯,π β² βͺ π‘ β² βͺ π’ β² β π₯,π β²β² = π₯ π β© π¦ β π₯ π , π‘ β²β² = π₯ π β© π§ β π₯ π , π’ β²β² = π₯ π β© π£ β π₯ π ,π β²β² βͺ π‘ β²β² βͺ π’ β²β² β π₯ π . (7.3) Ξ© ter ( π₯// X ) ter ( β // X ) ter ( π₯π¦// X ) ter ( π¦// X ) ter ( π₯π¦π§// X ) ter ( π₯π¦π£// X ) ter ( π¦π§// X ) ter ( π¦π£// X ) ter ( π₯π¦π§π£// X ) ter ( π¦π§π£// X ) ter ( π₯π§// X ) ter ( π₯π§π£// X ) ter ( π₯π£// X ) ter ( π§// X ) ter ( π§π£// X ) ter ( π£// X ) π¦ β© π₯ π¦ β© π₯ π π§ β© π₯ π£ β© π₯π§ β© π₯ π π£ β© π₯ π β β π₯ β β π₯ π Ξ© π β² π β© π‘ β² π β© π’ β² π β π₯π π β²β² π β© π‘ β²β² π β© π’ β²β² π β π₯π β²β© π‘ β² π β© π’ β² π π β²β²β© π‘ β²β² π β© π’ β²β² ππ β²β© π‘ β²β© π’ β² π π β²β© π‘ β² π β© π’ β² π β²β²β© π‘ β²β²β© π’ β²β² π π β²β²β© π‘ β²β² π β© π’ β²β² π β²β© π‘ β²β© π’ β² π β²β²β© π‘ β²β²β© π’ β²β² π β² π β© π‘ β²β© π’ β² π π β² π β© π‘ β²β© π’ β² π β² π β© π‘ β² π β© π’ β² π β²β² π β© π‘ β²β²β© π’ β²β² π π β²β² π β© π‘ β²β²β© π’ β²β² π β²β² π β© π‘ β²β² π β© π’ β²β² π β² π β²β² π‘ β² π’ β² π‘ β²β² π’ β²β² β β π₯ β β π₯ π Figure 11: Venn diagrams of the frame half-rare quadruplet of events X = { π₯, π¦, π§, π£ } , / (cid:62) π π₯ (cid:62) π π¦ (cid:62) π π§ (cid:62) π π£ (up), and two insertedtriplets π³ β² = { π β² , π‘ β² , π’ β² } and π³ β²β² = { π β²β² , π‘ β²β² , π’ β²β² } (down), agreed with theframe quadruplet X in the following sense: π¦ = π β² + π β²β² , π§ = π‘ β² + π‘ β²β² , π£ = π’ β² + π’ β²β² π β² βͺ π‘ β² βͺ π’ β² β π₯, π β²β² βͺ π‘ β²β² βͺ π’ β²β² β π₯ π . The recurrent formulas, which express the e.p.d.of the 1st kind of Λ π -ordered half-rare quadruplet ofevents X through the e.p.d. of the 1st kind of twoinserted triplets π³ β² and π³ β²β² , follow from the gen-eral recurrent formulas (3.52) and (3.53) in Note15 as well as in cases of a doublet and a triplet ofevents. And therefore, and because of the cumber-someness, these formulas are not represented here,but are only illustrated by Venn diagrams (see Fig.11). Let us dwell in more detail on FrΒ΄echet-restrictionsfor
11 = 2 β β functional parameters of a Kop-ula of quadruplet of events, to derive the recur-rent sequence of such FrΒ΄echet-restrictions, which begins with FrΒ΄echet-restrictions for a doublet ofevents (5.36), continues with FrΒ΄echet-restrictionsfor a triplet of events (6.16), and should be sup-ported by FrΒ΄echet-restrictions for parameters of aKopula of quadruplet of events X = { π₯, π¦, π§, π£ } andso on.To this end, we ο¬rst recall FrΒ΄echet-restrictions forparameters of Kopulas of a doublet and a triplet ofevents. For a Kopula of doublet of events, the FrΒ΄echet-restrictions of a β β parameter of insertedmonoplets π³ β² and π³ β²β² have the form: (cid:54) π π β² (cid:54) π π¦ . (7.4) For a Kopula of doublet of events, the FrΒ΄echet-restrictions of β β parameters of inserteddoublets π³ β² and π³ β²β² have the form: (cid:54) π π β² (cid:54) π π¦ , (cid:54) π π‘ β² (cid:54) π π§ ,π β π β² π‘ β² (cid:54) π π β² π‘ β² (cid:54) π + π β² π‘ β² ,π β π β²β² π‘ β²β² (cid:54) π π β²β² π‘ β²β² (cid:54) π + π β²β² π‘ β²β² , (7.5)where π β π β² π‘ β² = max { , π π β² + π π‘ β² β π π₯ } ,π + π β² π‘ β² = min { π π β² , π π‘ β² } ,π β π β²β² π‘ β²β² = max { , π π₯ + π π¦ + π π§ β β π π β² β π π‘ β² } ,π + π β²β² π‘ β²β² = min { π π¦ β π π β² , π π§ β π π‘ β² } , (7.6)are the lower and upper FrΒ΄echet-boundaries prob-abilities of double intersections of events from in-serted doublets π³ β² and π³ β²β² with respect to the framemonoplet { π₯ } .The case of triplet of events gives a newlevel of FrΒ΄echet-restrictions (the two last FrΒ΄echet-boundaries in (7.6)), when probabilities of dou-ble intersections of events from inserted doubletshave FrΒ΄echet-boundaries that depend not only onmarginal probabilities of the triplet, but and on in-serted marginal probabilities on which, in turn, theusual FrΒ΄echet-restrictions mentioned above are im-posed. For a Kopula of quadruplet of events the FrΒ΄echet-restrictions of
11 = 2 β β parameters of the in- OROBYEV serted triplet π³ β² and π³ β²β² have the form: (cid:54) π π β² (cid:54) π π¦ , (cid:54) π π‘ β² (cid:54) π π§ , (cid:54) π π’ β² (cid:54) π π£ ,π β π β² π‘ β² (cid:54) π π β² π‘ β² (cid:54) π + π β² π‘ β² ,π β π β² π’ β² (cid:54) π π β² π’ β² (cid:54) π + π β² π’ β² ,π β π‘ β² π’ β² (cid:54) π π‘ β² π’ β² (cid:54) π + π‘ β² π’ β² ,π β π β²β² π‘ β²β² (cid:54) π π β²β² π‘ β²β² (cid:54) π + π β²β² π‘ β²β² ,π β π β²β² π’ β²β² (cid:54) π π β²β² π’ β²β² (cid:54) π + π β²β² π’ β²β² ,π β π‘ β²β² π’ β²β² (cid:54) π π‘ β²β² π’ β²β² (cid:54) π + π‘ β²β² π’ β²β² ,π β π β² π‘ β² π’ β² (cid:54) π π β² π‘ β² π’ β² (cid:54) π + π β² π‘ β² π’ β² ,π β π β²β² π‘ β²β² π’ β²β² (cid:54) π π β²β² π‘ β²β² π’ β²β² (cid:54) π + π β²β² π‘ β²β² π’ β²β² , (7.7)where π β π β² π‘ β² = max { , π π β² + π π‘ β² β π π₯ } ,π + π β² π‘ β² = min { π π β² , π π‘ β² } ,π β π β²β² π‘ β²β² = max { , π π₯ + π π¦ + π π§ β β π π β² β π π‘ β² } ,π + π β²β² π‘ β²β² = min { π π¦ β π π β² , π π§ β π π‘ β² } ,π β π β² π’ β² = max { , π π β² + π π’ β² β π π₯ } ,π + π β² π’ β² = min { π π β² , π π’ β² } ,π β π β²β² π’ β²β² = max { , π π₯ + π π¦ + π π£ β β π π β² β π π’ β² } ,π + π β²β² π’ β²β² = min { π π¦ β π π β² , π π£ β π π’ β² } ,π β π‘ β² π’ β² = max { , π π‘ β² + π π’ β² β π π₯ } ,π + π‘ β² π’ β² = min { π π‘ β² , π π’ β² } ,π β π‘ β²β² π’ β²β² = max { , π π₯ + π π§ + π π£ β β π π‘ β² β π π’ β² } ,π + π‘ β²β² π’ β²β² = min { π π§ β π π‘ β² , π π£ β π π’ β² } ,π β π β² π‘ β² π’ β² = max { , π π β² π‘ β² + π π β² π’ β² + π π‘ β² π’ β² β π π₯ } ,π + π β² π‘ β² π’ β² = min { π π β² π‘ β² , π π β² π’ β² , π π‘ β² π’ β² } ,π β π β²β² π‘ β²β² π’ β²β² = max { , π π β²β² π‘ β²β² + π π β²β² π’ β²β² + π π‘ β²β² π’ β²β² β β π π₯ ) } ,π + π β²β² π‘ β²β² π’ β²β² = min { π π β²β² π‘ β²β² , π π β²β² π’ β²β² , π π‘ β²β² π’ β²β² } (7.8)are the lower and upper FrΒ΄echet-boundaries ofprobabilities of double and triple intersections ofevents from the inserted triplets π³ β² and π³ β²β² with re-spect to the frame monoplet { π₯ } .The case of a quadruplet of events gives the fol-lowing level of FrΒ΄echet-restrictions (the four lastFrΒ΄echet-boundaries in (7.8)), when probabilities oftriple intersections of events from inserted tripletshave FrΒ΄echet-boundaries that depend directly notso much on marginal probabilities as on insertedprobabilities of double intersections, on which, inturn, Frβechet-restrictions of the previous level,mentioned above, are imposed.The all FrΒ΄echet-restrictions in the considered framemethods for a doublet, a triplet and a quadru-plet of events differ from the usual FrΒ΄echet-restrictions, which are functions of only corre-sponding marginal probabilities. They differ in that they have a recurrent structure. When, as thepower of intersections of inserted events increases,the FrΒ΄echet-boundaries of their probabilities arefunctions of FrΒ΄echet-boundaries for probabilities ofintersections of lower power.The such FrΒ΄echet-restrictions and FrΒ΄echet-boundaries for a doublet (7.4), a triplet (7.5,7.6), aquadruplet (7.7,7.8) of events and so on, will callthe recurrent FrΒ΄echet-restrictions and recurrentFrΒ΄echet-boundaries . The recurrent formulas for Kopula of a quadru-plet of events immediately can be written in theform of special cases of recurrence formulas (3.52)and (3.53) from Note 15 for the quadruplet X = { π₯, π¦, π§, π£ } = { π₯ } + { π¦, π§, π£ } = { π₯ } + π³ : K (οΈ Λ π ( π | π + π// π³ + { π₯ } ) )οΈ == β§βͺβ¨βͺβ© K ( { π₯ } ) (οΈ Λ π ( π | π ( β©{ π₯ } // { π₯ } ) // π³ β² ) )οΈ , π = { π₯ } , K ( β ) (οΈ Λ π ( π | π ( β©β // { π₯ } ) // π³ β²β² ) )οΈ , π = β , (7.9) K (οΈ Λ π ( π | π + π// π³ + { π₯ } ) )οΈ == β§βͺβ¨βͺβ© K |{ π₯ } (οΈ Λ π ( π | π//
π³ |{ π₯ } // { π₯ } ) )οΈ π π₯ , π = { π₯ } , K |β (οΈ Λ π ( π | π//
π³ |β // { π₯ } ) )οΈ (1 β π π₯ ) , π = β . (7.10)In the formulas (7.9) the inserted pseudo-Kopulas K ( { π₯ } ) and K ( β ) are deο¬ned by the octuples of prob-abilities, i.e., by the formulas: K ( { π₯ } ) (οΈ Λ π ( π | π ( β©{ π₯ } // { π₯ } ) // π³ β² ) )οΈ == β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© π π β² π‘ β² π’ β² , π = { π¦, π§, π£ } ,π π β² π‘ β² β π π β² π‘ β² π’ β² , π = { π¦, π§ } ,π π β² π’ β² β π π β² π‘ β² π’ β² , π = { π¦, π£ } ,π π‘ β² π’ β² β π π β² π‘ β² π’ β² , π = { π§, π£ } ,π π β² β π π β² π‘ β² β π π β² π’ β² + π π β² π‘ β² π’ β² , π = { π§ } ,π π‘ β² β π π β² π‘ β² β π π‘ β² π’ β² + π π β² π‘ β² π’ β² , π = { π£ } ,π π’ β² β π π β² π’ β² β π π‘ β² π’ β² + π π β² π‘ β² π’ β² , π = { π£ } ,π π₯ β π π β² β π π‘ β² β π π’ β² ++ π π β² π‘ β² + π π β² π’ β² + π π‘ β² π’ β² β π π β² π‘ β² π’ β² , π = β , (7.11)
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ONFERENCE K ( β ) (οΈ Λ π ( π | π ( β©β // { π₯ } ) // π³ β²β² ) )οΈ == β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© π π β²β² π‘ β²β² π’ β²β² , π = { π¦, π§, π£ } ,π π β²β² π‘ β²β² β π π β²β² π‘ β²β² π’ β²β² , π = { π¦, π§ } ,π π β²β² π’ β²β² β π π β²β² π‘ β²β² π’ β²β² , π = { π¦, π£ } ,π π‘ β²β² π’ β²β² β π π β²β² π‘ β²β² π’ β²β² , π = { π§, π£ } ,π π¦ β π π β² ββ π π β²β² π‘ β²β² β π π β²β² π’ β²β² + π π β²β² π‘ β²β² π’ β²β² , π = { π¦ } ,π π§ β π π‘ β² ββ π π β²β² π‘ β²β² β π π‘ β²β² π’ β²β² + π π β²β² π‘ β²β² π’ β²β² , π = { π§ } ,π π£ β π π‘ β² ββ π π β²β² π’ β²β² β π π‘ β²β² π’ β²β² + π π β²β² π‘ β²β² π’ β²β² , π = { π£ } , β π π₯ β π π¦ β π π§ β π π£ ++ π π β² + π π‘ β² + π π’ β² ββ π π β²β² π‘ β²β² β π π β²β² π’ β²β² β π π‘ β²β² π’ β²β² ++ π π β²β² π‘ β²β² π’ β²β² , π = β , (7.12)where, for example, for π = { π¦, π§, π£ }{ π¦, π§, π£ } ( β©{ π₯ } // { π₯ } ) = { π¦ β© π₯, π§ β© π₯, π£ β© π₯ } β π³ β² , { π¦, π§, π£ } ( β©β // { π₯ } ) = { π¦ β© π₯ π , π§ β© π₯ π , π£ β© π₯ π } β π³ β²β² , (7.13)and the corresponding sets of marginal probabili-ties of inserted triplets π³ β² and π³ β²β² have the form Λ π ( π |{ π¦,π§,π£ } ( β©{ π₯ } // { π₯ } ) // π³ β²β² ) = { π π β² , π π‘ β² , π π’ β² } , Λ π ( π |{ π¦,π§,π£ } ( β©β // { π₯ } ) // π³ β²β² ) = { π π¦ β π π β² , π π§ β π π‘ β² , π π£ β π π’ β² } . (7.14)In the formulas (7.10) the condirional Kopulas K |{ π₯ } and K |β are deο¬ned by the octuples of proba-bilities that are normalized by π π₯ and by β π π₯ cor-respondingly, i.e., by the formulas: K |{ π₯ } (οΈ Λ π ( π | π//
π³ |{ π₯ } // { π₯ } ) )οΈ == β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© π π₯ π π β² π‘ β² π’ β² , π = { π¦, π§, π£ } , π π₯ ( π π β² π‘ β² β π π β² π‘ β² π’ β² ) , π = { π¦, π§ } , π π₯ ( π π β² π’ β² β π π β² π‘ β² π’ β² ) , π = { π¦, π£ } , π π₯ ( π π‘ β² π’ β² β π π β² π‘ β² π’ β² ) , π = { π§, π£ } , π π₯ ( π π β² β π π β² π‘ β² β π π β² π’ β² + π π β² π‘ β² π’ β² ) , π = { π§ } , π π₯ ( π π‘ β² β π π β² π‘ β² β π π‘ β² π’ β² + π π β² π‘ β² π’ β² ) , π = { π£ } , π π₯ ( π π’ β² β π π β² π’ β² β π π‘ β² π’ β² + π π β² π‘ β² π’ β² ) ,π = { π£ } , β π π₯ ( π π β² + π π‘ β² + π π’ β² )++ π π₯ ( π π β² π‘ β² + π π β² π’ β² + π π‘ β² π’ β² ) ββ π π₯ π π β² π‘ β² π’ β² , π = β , (7.15) K |β (οΈ Λ π ( π | π//
π³ |β // { π₯ } ) )οΈ == β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© β π π₯ π π β²β² π‘ β²β² π’ β²β² , π = { π¦, π§, π£ } , β π π₯ ( π π β²β² π‘ β²β² β π π β²β² π‘ β²β² π’ β²β² ) , π = { π¦, π§ } , β π π₯ ( π π β²β² π’ β²β² β π π β²β² π‘ β²β² π’ β²β² ) , π = { π¦, π£ } , β π π₯ ( π π‘ β²β² π’ β²β² β π π β²β² π‘ β²β² π’ β²β² ) , π = { π§, π£ } , β π π₯ ( π π¦ β π π β² ) ββ β π π₯ ( π π β²β² π‘ β²β² + π π β²β² π’ β²β² β π π β²β² π‘ β²β² π’ β²β² ) , π = { π¦ } , β π π₯ ( π π§ β π π‘ β² ) ββ β π π₯ ( π π β²β² π‘ β²β² + π π‘ β²β² π’ β²β² β π π β²β² π‘ β²β² π’ β²β² ) , π = { π§ } , β π π₯ ( π π£ β π π‘ β² ) ββ β π π₯ ( π π β²β² π’ β²β² + π π‘ β²β² π’ β²β² β π π β²β² π‘ β²β² π’ β²β² ) , π = { π£ } , β β π π₯ ( π π¦ + π π§ + π π£ )++ β π π₯ ( π π β² + π π‘ β² + π π’ β² ) ββ β π π₯ ( π π β²β² π‘ β²β² + π π β²β² π’ β²β² + π π‘ β²β² π’ β²β² )++ β π π₯ π π β²β² π‘ β²β² π’ β²β² , π = β , (7.16)The corresponding sets of marginal conditionalprobabilities of events π¦, π§ β π³ with respect tothe frame terraced events ter ( { π₯ } // { π₯ } ) = π₯ and ter ( β // { π₯ } ) = π₯ π correspondingly have the form: Λ π ( π |π³ // π³ |{ π₯ } // { π₯ } ) = {οΈ π π β² π π₯ , π π‘ β² π π₯ , π π’ β² π π₯ }οΈ , Λ π ( π |π³ // π³ |β // { π₯ } ) = {οΈ π π¦ β π π β² β π π₯ , π π§ β π π‘ β² β π π₯ , π π£ β π π’ β² β π π₯ }οΈ . (7.17)Recall that 11 functional parameters π π β² , π π‘ β² , π π’ β² ,π π β² π‘ β² , π π β² π’ β² , π π‘ β² π’ β² ,π π β²β² π‘ β²β² , π π β²β² π’ β²β² , π π‘ β²β² π’ β²β² ,π π β² π‘ β² π’ β² , π π β²β² π‘ β²β² π’ β²β² (7.18)in the recurrent formulas (7.11, 7.12) and (7.15,7.16) obey the FrΒ΄echet-restrictions (7.7) andFrΒ΄echet-boundaries (7.8). π -Kopula First, without the frame method, which is not re-quired here, letβs consider the simplest exampleof the π -Kopula K β Ξ¨ X of an π -set of events X ,i.e., a 1-function on the unit X -hypercube. In otherwords, we construct a nonnegative bounded nu-merical function K : [0 , β X β [0 , , that for all π§ β X βοΈ π₯ β π β X K (οΈ Λ π€ ( π | π// X ) )οΈ = π€ π₯ . A such simplest example of a 1-function on the unit X -hypercube is the so-called independent π -Kopula OROBYEV which for all free variables Λ π€ β [0 , β X is deο¬nedby the formula: K ( Λ π€ ) = βοΈ π₯ β X π€ π₯ , (8.1)that provides it on each ( π | Λ π€ ) -phenomenon-domthe following π values: K (οΈ Λ π€ ( π | π// X ) )οΈ = βοΈ π₯ β π π€ π₯ βοΈ π₯ β X β π (1 β π€ π₯ ) (8.2)for π β X . Indeed as in the case of doublet ofevents this function is a 1-function, since for all π₯ β X βοΈ π₯ β π β X (οΈ βοΈ π§ β π π€ π§ βοΈ π§ β X β π (1 β π€ π§ ) )οΈ = π€ π₯ . The e.p.d. of the 1st kind of ondependent π -s.e. X with the X -set of probabilities of events Λ π is deο¬nedby π values of the independent π -Kopula (8.1) onthe ( π | Λ π ) -phenomenon-dom by the general formu-las of half-rare variables, i.e., for π β { π₯, π¦ } : π ( π// X ) = K (οΈ Λ π ( π | π// X ) )οΈ = βοΈ π₯ β π π π₯ βοΈ π₯ β X β π (1 β π π₯ ) . (8.3) The general recurrent formulas (3.52, 3.53) of theframe method for constructing a Kopula of a set ofevents are derived in Note 15. Recall these formu-las: K (οΈ Λ π ( π | π + π// π³ + { π₯ } ) )οΈ == β§βͺβ¨βͺβ© K ( { π₯ } ) (οΈ Λ π ( π | π ( β©{ π₯ } // { π₯ } ) // π³ β² ) )οΈ , π = { π₯ } , K ( β ) (οΈ Λ π ( π | π ( β©β // { π₯ } ) // π³ β²β² ) )οΈ , π = β , (8.4) K (οΈ Λ π ( π | π + π// π³ + { π₯ } ) )οΈ == β§βͺβ¨βͺβ© K |{ π₯ } (οΈ Λ π ( π | π//
π³ |{ π₯ } // { π₯ } ) )οΈ π , π = { π₯ } , K |β (οΈ Λ π ( π | π//
π³ |β // { π₯ } ) )οΈ (1 β π ) , π = β , (8.5)which express the π -Kopula of π -s.e. X = π³ + { π₯ } ,where π³ = π³ β² (+) π³ β²β² , through the known probabil-ity π of the event π₯ and together with it eithertwo known inserted pseudo- ( π β -Kopulas (see Def-inition 9), i.e., pseudo- ( π β -Kopulas of inserted ( π β -s.e.βs π³ β² and π³ β²β² in the frame monoplet { π₯ } ,or through two known conditional ( π β -Kopulas (see Deο¬nition 10) with respect to the frame mono-plet { π₯ } for the same π³ β² and π³ β²β² . Perhaps this statement deserves to be called a lemma,which, incidentally, is not diο¬cult to prove.
We write out more detailed formulas for corre-sponding pseudo- ( π β -Kopulas: K ( { π₯ } ) (οΈ Λ π ( π | π ( β©{ π₯ } // { π₯ } ) // π³ β² ) )οΈ = K ( { π₯ } ) (οΈ Λ π ( π | π β² // π³ β² ) )οΈ == {οΈ π ( π β² // π³ β² ) , π β² ΜΈ = β ,π ( β // π³ β² ) β π , π β² = β , K ( β ) (οΈ Λ π ( π | π ( β©β // { π₯ } ) // π³ β²β² ) )οΈ = K ( β ) (οΈ Λ π ( π | π β²β² // π³ β²β² ) )οΈ == {οΈ π ( π β²β² // π³ β²β² ) , π β²β² ΜΈ = β ,π ( β // π³ β²β² ) β π , π β²β² = β , (8.6)and for conditional ( π β -Kopulas: K |{ π₯ } (οΈ Λ π ( π | π//
π³ |{ π₯ } // { π₯ } ) )οΈ == {οΈ π π ( π β² // π³ β² ) , π β² ΜΈ = β , π ( π ( β // π³ β² ) β π ) , π β² = β , K |β (οΈ Λ π ( π | π//
π³ |β // { π₯ } ) )οΈ == {οΈ β π π ( π β²β² // π³ β²β² ) , π β²β² ΜΈ = β , β π ( π ( β // π³ β²β² ) β π ) , π β²β² = β , (8.7)where π β² = π ( β©{ π₯ } // { π₯ } ) = π ( β© ) { π₯ } = { π₯ β© π₯ , π₯ β π } π β²β² = π ( β©β // { π₯ } ) = π ( β© ) { π₯ π } = { π₯ β© π₯ π , π₯ β π } π β π³ . Now let us consider recurrent formulas forFrΒ΄echet-boundaries FrΒ΄echet-restrictions and forthe π β π β functional parameters of an π -Kopulaof π -set of events X = { π₯ , π₯ , . . . , π₯ π β } == { π₯ } + { π₯ , . . . , π₯ π β } == { π₯ } + π³ == { π₯ } + ( π³ β² (+) π³ β²β² ) , (8.8)where π³ β² = { π₯ β© π₯ , . . . , π₯ β© π₯ π β } , π³ β²β² = { π₯ π β© π₯ , . . . , π₯ π β© π₯ π β } (8.9)are inserted ( π β -s.e.βs, and Λ π ( π | X // X ) = { π , π , . . . , π π β } (8.10)is the X -set of probabilities of marginal events from X , i.e., π π = P ( π₯ π ) , π = 0 , , . . . , π β .Judging by the form of FrΒ΄echet-boundariesFrΒ΄echet-restrictions for a doublet, a triplet and
06 T HE XIV FAMEMSβ2015 C
ONFERENCE a quadruplet of events, collected in paragraph 7.2,these FrΒ΄echet-restrictions consists of two groups,such that one of them, which refers to the parame-ters of the inserted ( π β -s.e. π³ β² , consists of π β β FrΒ΄echet-restrictions, and the other, which refersto the parameters of the inserted ( π β -s.e. π³ β²β² ,consists of π β β ( π β β FrΒ΄echet-restrictions.And, as it should: π β π β π β β π β β ( π β β . (8.11)The ο¬rst group, related to the inserted ( π β -s.e. π³ β² , contains FrΒ΄echet-restrictions for probabilitiesof the second kind π π β² // π³ β² = P (οΈ βοΈ π₯ β² β π β² π₯ β² )οΈ , (8.12)that are numbered by nonempty subsets π β² ΜΈ = β ofinserted ( π β -s.e. π³ β² (the number od such subsets: π β β ); the second group, related to the inserted ( π β -s.e. π³ β²β² , contains FrΒ΄echet-restrictions for thesuch probabilities of the second kind: π π β²β² // π³ β²β² = P (οΈ βοΈ π₯ β²β² β π β²β² π₯ β²β² )οΈ , (8.13)that are numbered by subsets π β²β² β π³ β²β² with thepower | π β²β² | (cid:62) (number of such subsets: π β β ( π β β ). Note 18 (denotations for subsets of fixed power).
To more conveniently represent the recurrentFrΒ΄echet-restrictions, agree to denote π β² π β π³ β² ββ | π β² π | = π,π β²β² π β π³ β²β² ββ | π β²β² π | = π. (8.14)the subsets consisting of π events. In this nota-tion, for example, the X βset of marginal probabil-ities Λ π ( π | X // X ) is written as the X -set probabilities ofthe second kind that are numbered by monopletsof events π = { π₯ } , π₯ β X : { π , π , . . . , π π β } = { π π // X , π β X } . (8.15)The set of probabilities of double intersection ofevents π₯ β X , i.e., the set of probabilities of the sec-ond kind that are numbered by doublets, has theform: { π { π₯,π¦ } // X , { π₯, π¦ } β X } = { π π // X , π β X } . (8.16)And the set of probabilities of triple intersectionsof events π₯ β X , i.e., the set of probabilities of thesecond kind that are numbered by triplets, has theform: { π { π₯,π¦,π§ } // X , { π₯, π¦, π§ } β X } = { π π // X , π β X } (8.17)and so on. Note 19 (recurrent formulas for FrΒ΄echet-boundaries and FrΒ΄echet-restrictions).
Probabilitiesof π -intersections ( π = 2 , ..., π β of events from theinserted s.e.βs π³ β² and π³ β²β² have the recurrent FrΒ΄echet-restrictions (see paragraph 7.2) that are writtenby denotations from Note 18 by the following waywith respect to π³ β² : π β π β² π // π³ β² (cid:54) π π β² π // π³ β² (cid:54) π + π β² π // π³ β² , (8.18)where π β π β² π // π³ β² = max β§β¨β© , π π₯ β βοΈ π β² π β β π β² π ( π π₯ β π π β² π β // π³ β² ) β«β¬β ,π + π β² π // π³ β² = min π β² π β β π β² π {οΈ π π β² π β // π³ β² , π β² π β β π³ β² }οΈ . (8.19)are recurrent the lower and upper FrΒ΄echet-boundaries . And the lower FrΒ΄echet-boundarywe can write somewhat differently after simpletransformations: π β π β² π // π³ β² = max β§β¨β© , βοΈ π β² π β β π β² π π π β² π β // π³ β² β ( π β π π₯ β«β¬β . (8.20)Similar look the recurrent FrΒ΄echet-restrictions withrespect to the inserted s.e. π³ β²β² : π β π β²β² π // π³ β²β² (cid:54) π π β²β² π // π³ β²β² (cid:54) π + π β²β² π // π³ β²β² , (8.21)where π β π β²β² π // π³ β²β² = max β§β¨β© , β π π₯ β βοΈ π β²β² π β β π β²β² π (1 β π π₯ β π π β²β² π β // π³ β²β² ) β«β¬β ,π + π β²β² π // π³ β²β² = min π β²β² π β β π β²β² π {οΈ π π β²β² π β // π³ β²β² , π β²β² π β β π³ β²β² }οΈ . (8.22)are recurrent the lower and upper FrΒ΄echet-boundaries . And the lower FrΒ΄echet-boundarywe can write somewhat differently after simpletransformations: π β π β²β² π // π³ β²β² == max β§β¨β© , βοΈ π β²β² π β β π β²β² π π π β²β² π β // π³ β²β² β ( π β β π π₯ ) β«β¬β . (8.23)It remains to write out more π β recurrent FrΒ΄echet-restrictions on probabilities of marginal eventsfrom the inserted s.e. π³ β² , i.e., on probabilities ofthe second kind that are numbered by monoplets π β² β π³ β² : (cid:54) π π β² // π³ β² (cid:54) π π // π³ , (8.24)which are restricted by marginal probabilities ofevents from the ( π β -s.e. π³ and which to-gether with the recurrent FrΒ΄echet-restrictions (8.18, OROBYEV recurrent FrΒ΄echet-restrictions . This totality consists of π β π β restric-tions. And recurrent the lower and upper FrΒ΄echet-boundaries in these restrictions are deο¬ned by re-current formulas (8.19, 8.22). Letβs consider parametrization on an example offunctional parameters π π β² , π π‘ β² , π π β² π‘ β² π π β²β² π‘ β²β² of 3-Kopulaof the Λ π -ordered half-rare triplet X = { π₯, π¦, π§ } , whichin the frame method is constructed from two in-serted pseudo-2-Kopulas. π π β² and π π‘ β² The FrΒ΄echet-restriction of the functional parameter π π β² = π π β² ( π π₯ , π π¦ , π π§ ) 0 (cid:54) π π β² (cid:54) π π¦ , (9.1)that in the frame method has a sense of probabilityof double intersection of events π₯ and π¦ : π π β² = π π₯π¦// X = P ( π₯ β© π¦ ) , (9.2)is baced on the notion of FrΒ΄echet-correlation [1] Kor π₯π¦ = β§βͺβ¨βͺβ© Kov π₯π¦ | Kov β π₯π¦ | , Kov π₯π¦ < , Kov π₯π¦ Kov + π₯π¦ , Kov π₯π¦ (cid:62) , (9.3)where Kov π₯π¦ = P ( π₯ β© π¦ ) β P ( π₯ ) P ( π¦ ) (9.4)is a covariance of events π₯ and π¦ , and Kov β π₯π¦ = max { , π π₯ + π π¦ β } β π π₯ π π¦ = β π π₯ π π¦ , Kov + π₯π¦ = min { π π₯ , π π¦ } β π π₯ π π¦ = π π¦ β π π₯ π π¦ (9.5)are its the lower and upper FrΒ΄echet-boundaries.From Deο¬nition (9.3) we get the parametrizationof functional parameter π π β² by the double FrΒ΄echet-correlation on the following form: π π β² ( π π₯ , π π¦ , π π§ ) == {οΈ π π₯ π π¦ β Kor π₯π¦ Kov β π₯π¦ , Kor π₯π¦ < ,π π₯ π π¦ + Kor π₯π¦ Kov + π₯π¦ , Kor π₯π¦ (cid:62) {οΈ π π₯ π π¦ + Kor π₯π¦ π π₯ π π¦ , Kor π₯π¦ < ,π π₯ π π¦ + Kor π₯π¦ ( π π¦ β π π₯ π π¦ ) , Kor π₯π¦ (cid:62) . (9.6) The parametrization of functional parameter π π‘ β² bythe double FrΒ΄echet-correlation is similar: π π‘ β² ( π π₯ , π π¦ , π π§ ) == {οΈ π π₯ π π§ β Kor π₯π§ Kov β π₯π§ , Kor π₯π§ < ,π π₯ π π§ + Kor π₯π§ Kov + π₯π§ , Kor π₯π§ (cid:62) {οΈ π π₯ π π§ + Kor π₯π§ π π₯ π π§ , Kor π₯π§ < ,π π₯ π π§ + Kor π₯π§ ( π π§ β π π₯ π π§ ) , Kor π₯π§ (cid:62) . (9.7) We recall ο¬rst that an absolute triple FrΒ΄echet-correlation [1] of three events π₯, π¦ and π§ is deο¬nedsimilarly to the double one: Kor π₯π¦π§ = β§βͺβ¨βͺβ© Kov π₯π¦π§ | Kov β π₯π¦π§ | , Kov π₯π¦π§ < , Kov π₯π¦π§
Kov + π₯π¦π§ , Kov π₯π¦π§ (cid:62) , (9.8)where Kov π₯π¦π§ = P ( π₯ β© π¦ β© π§ ) β P ( π₯ ) P ( π¦ ) P ( π§ ) (9.9)is the triple covariance of events π₯, π¦ and π§ , and Kov β π₯π¦π§ = max { , π π₯ + π π¦ + π π§ β } β π π₯ π π¦ π π§ == β π π₯ π π¦ π π§ , Kov + π₯π¦π§ = min { π π₯ , π π¦ , π π§ } β π π₯ π π¦ π π§ == π π§ β π π₯ π π¦ π π§ (9.10)are its absolute the lower and upper FrΒ΄echet-boundaries.The deο¬nition of the inserted triple FrΒ΄echet-correlation differs of the deο¬nition of absolute one (9.8) in that its the lower and upper FrΒ΄echet-boundaries must depend on the e.p.d. of the in-serted doublets π³ β² and π³ β²β² . So they differ from ab-solute FrΒ΄echet-boundaries (9.10) and have the form(9.19), where π β π β² π‘ β² = max { , π π β² + π π‘ β² β π π₯ } ,π + π β² π‘ β² = min { π π β² , π π‘ β² } ,π β π β²β² π‘ β²β² = max { , π π₯ + π π¦ + π π§ β β π π β² β π π‘ β² } ,π + π β²β² π‘ β²β² = min { π π¦ β π π β² , π π§ β π π‘ β² } , (9.11)are the lower and upper FrΒ΄echet-boundaries ofprobabilities of double intersections of events from inserted doublets π³ β² and π³ β²β² with respect to theframe monoplet { π₯ } , which should serve inserted the lower and upper FrΒ΄echet-boundaries of proba-bilities of triple intersections (9.20) of events fromthe triplet X = { π₯ } + ( π³ β² (+) π³ β²β² ) . However, asmight be expected, these FrΒ΄echet-boundaries arenot always ready to serve as the lower and upperFrΒ΄echet-boundaries for probabilities of triple inter-sections.
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For this reason, it is necessary to modify the deο¬-nitions of two inserted triple covariances and, re-spectively, β inserted the lower and upper FrΒ΄echet-boundaries of these covariances.
The ο¬rst modiο¬cation of deο¬nitions (see Fig. 18,19,20).
For brevity, we denote π ( { π₯ } ) β = π π₯ π π¦ π π§ , π ( β ) β = (1 β π π₯ ) π π¦ π π§ . Two inserted triple covariance are deο¬nedby the ο¬rmulas: Kov ( { π₯ } ) π₯π¦π§ = β§βͺβ¨βͺβ© π π β² π‘ β² β π ( { π₯ } ) β , π ( { π₯ } ) β β [οΈ π β π β² π‘ β² , π + π β² π‘ β² ]οΈ ,π β π β² π‘ β² β π ( { π₯ } ) β , π ( { π₯ } ) β < π β π β² π‘ β² ,π + π β² π‘ β² β π ( { π₯ } ) β , π + π β² π‘ β² < π ( { π₯ } ) β , Kov ( β ) π₯π¦π§ = β§βͺβ¨βͺβ© π π β²β² π‘ β²β² β π ( β ) β , π ( β ) β β [οΈ π β π β² π‘ β² , π + π β² π‘ β² ]οΈ ,π β π β²β² π‘ β²β² β π ( β ) β , π ( β ) β < π β π β²β² π‘ β²β² ,π + π β²β² π‘ β²β² β π ( β ) β , π + π β²β² π‘ β²β² < π ( β ) β , (9.12)and inserted the lower and upper FrΒ΄echet-boundaries of these covariances β by the for-mulas: Kov β ( { π₯ } ) π₯π¦π§ = β§βͺβ¨βͺβ© π β π β² π‘ β² β π ( { π₯ } ) β , π ( { π₯ } ) β β [οΈ π β π β² π‘ β² , π + π β² π‘ β² ]οΈ ,π β π β² π‘ β² β π ( { π₯ } ) β , π ( { π₯ } ) β < π β π β² π‘ β² ,π + π β² π‘ β² β π ( { π₯ } ) β , π + π β² π‘ β² < π ( { π₯ } ) β , Kov +( { π₯ } ) π₯π¦π§ = β§βͺβ¨βͺβ© π + π β² π‘ β² β π ( { π₯ } ) β , π ( { π₯ } ) β β [οΈ π β π β² π‘ β² , π + π β² π‘ β² ]οΈ ,π β π β² π‘ β² β π ( { π₯ } ) β , π ( { π₯ } ) β < π β π β² π‘ β² ,π + π β² π‘ β² β π ( { π₯ } ) β , π + π β² π‘ β² < π ( { π₯ } ) β , (9.13) Kov β ( β ) π₯π¦π§ = β§βͺβ¨βͺβ© π β π β²β² π‘ β²β² β π ( β ) β , π ( β ) β β [οΈ π β π β² π‘ β² , π + π β² π‘ β² ]οΈ ,π β π β²β² π‘ β²β² β π ( β ) β , π ( β ) β < π β π β²β² π‘ β²β² ,π + π β²β² π‘ β²β² β π ( β ) β , π + π β²β² π‘ β²β² < π ( β ) β , Kov +( β ) π₯π¦π§ = β§βͺβ¨βͺβ© π + π β²β² π‘ β²β² β π ( β ) β , π ( β ) β β [οΈ π β π β² π‘ β² , π + π β² π‘ β² ]οΈ ,π β π β²β² π‘ β²β² β π ( β ) β , π ( β ) β < π β π β²β² π‘ β²β² ,π + π β²β² π‘ β²β² β π ( β ) β , π + π β²β² π‘ β²β² < π ( β ) β , (9.14)We introduce some notation for brevity of the for-mulas: π ( { π₯ } )0 = β§βͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβ© π { π₯ } β , if π { π₯ } β β [οΈ π + π β² π‘ β² , π β π β² π‘ β² ]οΈ ,π β π β² π‘ β² + ( π β π β² π‘ β² β π { π₯ } β )( π + π β² π‘ β² β π β π β² π‘ β² ) π + π β² π‘ β² + π β π β² π‘ β² β π ( { π₯ } ) β , if π { π₯ } β ΜΈβ [οΈ π + π β² π‘ β² , π β π β² π‘ β² ]οΈ ,π ( β )0 = β§βͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβ© π β β , π β β β [οΈ π + π β² π‘ β² , π β π β² π‘ β² ]οΈ ,π β π β²β² π‘ β²β² + ( π β π β²β² π‘ β²β² β π ( β ) β )( π + π β² π‘ β² β π β π β² π‘ β² ) π + π β²β² π‘ β²β² + π β π β²β² π‘ β²β² β π ( β ) β ,π β β ΜΈβ [οΈ π + π β² π‘ β² , π β π β² π‘ β² ]οΈ , (9.15) Two inserted triple covariances are deο¬ned by theformulas: Kov ( { π₯ } ) π₯π¦π§ = π π β² π‘ β² β π ( { π₯ } )0 Kov ( β ) π₯π¦π§ = π π β²β² π‘ β²β² β π ( β )0 , (9.16)and inserted the lower and upper FrΒ΄echet-boundaries of these covariances β by the for-mulas: Kov β ( { π₯ } ) π₯π¦π§ = π β π β² π‘ β² β π ( { π₯ } )0 , Kov +( { π₯ } ) π₯π¦π§ = π + π β² π‘ β² β π ( { π₯ } )0 , Kov β ( β ) π₯π¦π§ = π β π β²β² π‘ β²β² β π ( β )0 , Kov +( β ) π₯π¦π§ = π + π β²β² π‘ β²β² β π ( β )0 . (9.17) β β β For any modiο¬cation deο¬nitions of two inserted
FrΒ΄echet-correlations look in the usual way:
Kor ( { π₯ } ) π₯π¦π§ = β§βͺβͺβ¨βͺβͺβ© Kov ( { π₯ } ) π₯π¦π§ βββ Kov β ( { π₯ } ) π₯π¦π§ βββ , Kov ( { π₯ } ) π₯π¦π§ < , Kov ( { π₯ } ) π₯π¦π§ Kov +( { π₯ } ) π₯π¦π§ , Kov ( { π₯ } ) π₯π¦π§ (cid:62) , Kor ( β ) π₯π¦π§ = β§βͺβͺβ¨βͺβͺβ© Kov ( β ) π₯π¦π§ βββ Kov β ( β ) π₯π¦π§ βββ , Kov ( β ) π₯π¦π§ < , Kov ( β ) π₯π¦π§ Kov +( β ) π₯π¦π§ , Kov ( β ) π₯π¦π§ (cid:62) . (9.18) π π β² π‘ β² and π π β²β² π‘ β²β² The FrΒ΄echet-restriction of two functional parame-ters π π β² π‘ β² = π π β² π‘ β² ( π π₯ , π π¦ , π π§ ) π π β²β² π‘ β²β² = π π β²β² π‘ β²β² ( π π₯ , π π¦ , π π§ ) π β π β² π‘ β² (cid:54) π π β² π‘ β² (cid:54) π + π β² π‘ β² ,π β π β²β² π‘ β²β² (cid:54) π π β²β² π‘ β²β² (cid:54) π + π β²β² π‘ β²β² , (9.19)that in the frame method have a sense of probabil-ities of triple intersections of events: π π β² π‘ β² = P ( π₯ β© π¦ β© π§ ) ,π π β²β² π‘ β²β² = P ( π₯ π β© π¦ β© π§ ) , (9.20)is based on the notion of the inserted triple FrΒ΄echet-correlation.From deο¬nitions (9.18) and (9.11) we get theparametrization of functional parameter π π β² π‘ β² of the inserted triple FrΒ΄echet-correlation Kor ( { π₯ } ) π₯π¦π§ in thefollowing form: π π β² π‘ β² ( π π₯ , π π¦ , π π§ ) == β§βͺβͺβͺβͺβ¨βͺβͺβͺβͺβ© π π₯ π π¦ π π§ β Kor ( { π₯ } ) π₯π¦π§ Kov β ( { π₯ } ) π₯π¦π§ , Kor ( { π₯ } ) π₯π¦π§ < ,π π₯ π π¦ π π§ + Kor ( { π₯ } ) π₯π¦π§ Kov +( { π₯ } ) π₯π¦π§ , Kor ( { π₯ } ) π₯π¦π§ (cid:62) , (9.21) OROBYEV
The parametrization of functional parameter π π β²β² π‘ β²β² of the inserted triple FrΒ΄echet-correlation Kor ( β ) π₯π¦π§ fol-lows from the same deο¬nitions (9.18) and (9.11): π π β²β² π‘ β²β² ( π π₯ , π π¦ , π π§ ) == β§βͺβͺβͺβͺβ¨βͺβͺβͺβͺβ© π π₯ π π¦ π π§ β Kor ( β ) π₯π¦π§ Kov β ( β ) π₯π¦π§ , Kor ( { π₯ } ) π₯π¦π§ < ,π π₯ π π¦ π π§ + Kor ( β ) π₯π¦π§ Kov +( β ) π₯π¦π§ , Kor ( { π₯ } ) π₯π¦π§ (cid:62) , (9.22) Note 20 (about parametrization of functional pa-rameters of 3-Kopula by FrΒ΄echet-correlations).
The parametrization of the four functional pa-rameters π π β² , π π‘ β² , π π β² π‘ β² and π π β²β² π‘ β²β² of 3-Kopula of the Λ π -ordered half-rare triplet X = { π₯, π¦, π§ } by twodouble FrΒ΄echet-correlations Kor π₯π¦ and Kor π₯π§ (9.3)and by two inserted triple FrΒ΄echet-correlations Kor ( { π₯ } ) π₯π¦π§ and Kor ( β ) π₯π¦π§ (9.18) has the following ad-vantages. Each of four FrΒ΄echet-correlations isa numerical characteristics of dependency ofevents with values from ο¬xed interval [ β , +1] .And these values clearly indicate the proxim-ity to FrΒ΄echet-boundaries and to indepedent3-Kopula. The value β β β indicates to the lowerFrΒ΄echet-boundary, the value β +1 β β to the upperFrΒ΄echet-boundary, and the value β β β to inde-pendent events. For example, the equality of allthese four FrΒ΄echet-correlations to zero determinesa family of independent 3-Kopulas. Advantages ofthe proposed idea of parametrization of functionalparameters of 3-Kopula are that β an each FrΒ΄echet-correlation can take arbitraryvalue from [ β , +1] without any connectionwith the values of the other three FrΒ΄echet-correlations; β the above parametrization algorithm for func-tional parameters of 3-Kopula extends to theparametrization of the functional parametersof π -Kopulas by inserted FrΒ΄echet-correlationsof higher orders .
10 Examples of Kopulas of somefamilies of sets of events
Consider in Fig.
12 a number of examples of 2-Kopulas of doublets of half-rare events X = { π₯, π¦ } and its set-phenomena X ( π | π₯ ) = { π₯, π¦ π } , X ( π | π¦ ) = { π₯ π , π¦ } , and X ( π | π₯π¦ ) = { π₯ π , π¦ π } , each of which is char-acterized by its own functional parameter P ( π₯ β© In each ο¬gure, below the graph, maps of these 2-Kopulason unit squares in conditional colors are shown too, where thewhite color corresponds to the points at which the probabilitiesof terraced events are 1/4. π¦ ) = π π₯π¦ ( π€ π₯ , π€ π¦ ) , lying within the FrΒ΄echet bound-aries: (cid:54) π π₯π¦ ( π€ π₯ , π€ π¦ ) (cid:54) min { π€ π₯ , π€ π¦ } . (10.1)Upper 2-Kopula of FrΒ΄echet (embedded): π π₯π¦ ( π€ π₯ , π€ π¦ ) = min { π€ π₯ , π€ π¦ } . (10.2)Independent 2-Kopula of FrΒ΄echet: π π₯π¦ ( π€ π₯ , π€ π¦ ) = π€ π₯ π€ π¦ . (10.3)Lower 2-Kopula of FrΒ΄echet (minimum-intersected): π π₯π¦ ( π€ π₯ , π€ π¦ ) = max { , π€ π₯ + π€ π¦ β } . (10.4)Half-independent 2-Kopula: π π₯π¦ ( π€ π₯ , π€ π¦ ) = π€ π₯ π€ π¦ / . (10.5)Half-embedded 2-Kopula: π π₯π¦ ( π€ π₯ , π€ π¦ ) = min { π€ π₯ , π€ π¦ } / . (10.6)Arbitrary-embedded 2-Kopula: π π₯π¦ ( π€ π₯ , π€ π¦ ) == min { π€ π₯ , π€ π¦ } (1 + sin(15( π€ π₯ β π€ π¦ ))) / . (10.7)Continuously-arbitrary-embedded 2-Kopula: π π₯π¦ ( π€ π₯ , π€ π¦ ) == π€ π₯ π€ π¦ + ( πΌ ( π€ π₯ , π€ π¦ ) β π€ π₯ π€ π¦ ) π½ ( π€ π₯ , π€ π¦ ) , (10.8)where πΌ ( π€ π₯ , π€ π¦ ) == min { π€ π₯ , π€ π¦ } (1 + sin(15( π€ π₯ β π€ π¦ ))) / ,π½ ( π€ π₯ , π€ π¦ ) = βοΈ (1 / β π€ π₯ )(1 / β π€ π¦ ) . (10.9) In Fig.s 13, 14, 16, 17, and 15 it is shown 2-Kopulasof doublets of half-rare events X = { π₯, π¦ } and its set-phenomena X ( π | π₯ ) , X ( π | π¦ ) , and X ( π | π₯π¦ ) , correspondingto some classical copulas.2-Kopula of Ali-Mikhail-Haq, π β [ β , : π π₯π¦ ( π€ π₯ , π€ π¦ ) == π€ π₯ π€ π¦ β π (1 β π€ π₯ )(1 β π€ π¦ ) . (10.10)
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Figure 12: Cartesian representations of 2-Kopulas of doublets of half-rareevents X = { π₯, π¦ } and its set-phenomena X ( π | π₯ ) , X ( π | π¦ ) X ( π | π₯π¦ ) corre-sponding to Frechet Kopula (from up to down): upper (embedded), inde-pendent, lower (minimum-intersected).Figure 13: Cartesian representations of 2-Kopulas of doublets of half-rareevents X = { π₯, π¦ } and its set-phenomena X ( π | π₯ ) , X ( π | π¦ ) X ( π | π₯π¦ ) corre-sponding to Ali-Mikhail-Haq Kopula (from up to down): from near-upper( π = 0 . ) through independent ( π = 0 ) to lower ( π = β . ). π β [ β , β ) β { } : π π₯π¦ ( π€ π₯ , π€ π¦ ) == [οΈ max {οΈ π€ β ππ₯ + π€ β ππ¦ β
1; 0 }οΈ]οΈ β /π . (10.11) Figure 14: Cartesian representations of 2-Kopulas of doublets of half-rareevents X = { π₯, π¦ } and its set-phenomena X ( π | π₯ ) , X ( π | π¦ ) X ( π | π₯π¦ ) corre-sponding to Clayton Kopula (from up to down): from near-upper ( π =6 . ) around pinked-independent ( π = 0 . , β . ) to lower ( π = β . ).Figure 15: Cartesian representations of 2-Kopulas of doublets of half-rareevents X = { π₯, π¦ } and its set-phenomena X ( π | π₯ ) , X ( π | π¦ ) X ( π | π₯π¦ ) corre-sponding to Joe Kopula (from up to down): from independent ( π = 1 . )to near-lower ( π = 6 . ). π β [1 , β ) : π π₯π¦ ( π€ π₯ , π€ π¦ ) == 1 β [οΈ (1 β π€ π₯ ) π +(1 β π€ π¦ ) π β (1 β π€ π₯ ) π (1 β π€ π¦ ) π ]οΈ /π . (10.12) OROBYEV X = { π₯, π¦ } and its set-phenomena X ( π | π₯ ) , X ( π | π¦ ) X ( π | π₯π¦ ) corre-sponding to Frank Kopula (from up to down): from near-upper ( π = 6 . )around pinked-independent ( π = 0 . , β . ) to near-lower ( π = β . ).Figure 17: Cartesian representations of 2-Kopulas of doublets of half-rareevents X = { π₯, π¦ } and its set-phenomena X ( π | π₯ ) , X ( π | π¦ ) X ( π | π₯π¦ ) cor-responding to Gumbel Kopula (from up to down): from inndependent( π = 1 . ) to near-lower ( π = 6 . ). π β R β { } : π π₯π¦ ( π€ π₯ , π€ π¦ ) == β π log [οΈ β ππ€ π₯ ) β β ππ€ π¦ ) β β π ) β ]οΈ . (10.13) π β [1 , β ) : π π₯π¦ ( π€ π₯ , π€ π¦ ) == exp [οΈ β (οΈ ( β log( π€ π₯ )) π + ( β log( π€ π¦ )) π )οΈ /π ]οΈ . (10.14) In Fig.βs
18, 19, and 20 it is shown 3-Kopulas oftriplets of half-rare events X = { π₯, π¦, π§ } , functionalparameters of which serve FrΒ΄echet-correlations inthe ο¬rst modiο¬cation of deο¬nitions (see paragraph9.2).
11 Appendix
Consider the universal probability space (β¦ , π (cid:102) , P ) and one of its subject-name realizations , a par-tial probability space (β¦ , π , P ) . The elements ofthe sigma-algebra π (cid:102) are the universal Kolmogorovevents π₯ (cid:102) β π (cid:102) , and the elements of the sigma-algebra π β events π₯ β π , which serve as names ofuniversal Kolmogorov events π₯ (cid:102) (see in details [8]).The notions that are relevant to a s.e. X β π forwhich it is convenient to use the following abbrevi-ations: X = { π₯ : π₯ β X } β a set of events (s.e.); Λ π = { π π₯ , π₯ β X } β an X -set of probabilities of events from X ; X ( π | π ) = X ( π | π// X ) = { π₯ : π₯ β π } + { π₯ π : π₯ β X β π } β an π -phenomenon of X , π β X ; X ( π | X ) = X ( π | X // X ) = X β an X -phenomenon of X equal to X ; Λ π ( π | π// X ) = { π π₯ , π₯ β π } + { β π π₯ , π₯ β X β π } β an X -set of probabilities of events from X ( π | π ) , π β X ; Λ π ( π | X // X ) = Λ π β an X -set of probabilities of events from X ( π | X ) equal to Λ π ; π ( π// X ) β a value of e.p.d. of the 1st kind of X for π β X ; K (οΈ Λ π ( π | π// X ) )οΈ β a value of the Kopula of e.p.d. of the 1st kind of X for π β X ; π ( π// X ) = K (οΈ Λ π ( π | π// X ) )οΈ β the deο¬nition of e.p.d. of the 1st kind of X by its Kopula, π β X ; Λ π = { π π₯ , π₯ β X } β [0 , / β X β an X -set of half-rare variables; Λ π€ = { π€ π₯ , π₯ β X } β [0 , β X β an X -set of free variables. Each ο¬gure shows maps of these 3-copulas on a cube in con-ditional colors, where the white color corresponds to the pointsat which the probabilities of terraced events are 1/8.
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Kor = β Kor = β . Kor = β . Kor = β . Kor = 0
Kor = 0 . Kor = 0 . Kor = 0 . Kor = 1
Figure 18: The ο¬rst modiο¬cation of deο¬nitions. Cartesian representationsof 3-Kopulas of triplets of half-rare events X = { π₯, π¦, π§ } , constructed bythe frame method (6.18) with non-negative values of a single parame-ter (from up to down) Kor = β , β . , β . , β . , , . , . , . , , towhich all four inserted Frechet-correlations are equal (see paragraph 9).The independent 3-Kopula is obtained for
Kor = 0 . Lemma 5 (Set-phenomenon renumbering a e.p.d. of the1st kind and its Kopulas).
E.p.d. of the 1st kind andKopulas of the s.e. X and of its π -phenomena X ( π | π ) are connected by formulas of mutually inversionset-phenomenon renumbering for π β X π β X : π (οΈ π ( π | π β© π ) // X ( π | π ) )οΈ = π (( π β π ) π // X ) ,π ( π// X ) = π (οΈ (( π β π ) π ) ( π | π β© ( π Ξ π ) π ) // X ( π | π ) )οΈ ; (11.1) Kor = β Kor = β . Kor = β . Kor = β . Kor = β . Kor = β . Kor = β . Kor = β . Kor = β . Kor = β . Kor = 0
Figure 19: The ο¬rst modiο¬cation of deο¬nitions. Cartesian representationsof 3-Kopulas of triplets of half-rare events X = { π₯, π¦, π§ } , constructed bythe frame method (6.18) with non-positive values of the parameter Kor = β , β . , ..., β . , β . , (from up to down), to which inserted Frechet-correlations are equal (see paragraph 9). The independent 3-Kopula isobtained for
Kor = 0 . K (οΈ Λ π ( π | π ( π | π β© π ) // X ( π | π ) ) )οΈ = K (οΈ Λ π ( π | ( π Ξ π ) π // X ) )οΈ , K (οΈ Λ π ( π | π// X ) )οΈ = K (οΈ Λ π ( π | (( π Ξ π ) π ) ( π | π β© ( π Ξ π ) π ) // X ( π | π ) ) )οΈ . (11.2) OROBYEV
Kor = 0
Kor = 0 . Kor = 0 . Kor = 0 . Kor = 0 . Kor = 0 . Kor = 0 . Kor = 0 . Kor = 0 . Kor = 0 . Kor = 1
Figure 20: The ο¬rst modiο¬cation of deο¬nitions. Cartesian representationsof 3-Kopulas of triplets of half-rare events X = { π₯, π¦, π§ } , constructedby the frame method (6.18) with non-negative values of the parameter Kor = 0 , . , . , ..., . , (from up to down), to which inserted Frechet-correlations are equal (see paragraph 9). The independent 3-Kopula isobtained for
Kor = 0 . Proof follows immediately from the formulas ofthe set-phenomenon renumbering the terraced events of the 1st kind of the s.e. and of its set-phenomena proved in [6]. Λ π -order The following special denotations for a doublet ofevents X = { π₯, π¦ } and the X -set of marginal proba-bilities Λ π = { π π₯ , π π¦ } that are invariant relative to the Λ π -order, are useful. X = { π₯, π¦ } = { π₯ β , π₯ β } , Λ π = { π π₯ , π π¦ } = { π β , π β } , / (cid:62) π β (cid:62) π β , Λ π€ = { π€ π₯ , π€ π¦ } = { π€ β , π€ β } , (cid:62) π€ β (cid:62) π€ β , (11.3) { π₯ β } = max { X } = {οΈ { π₯ } , π π₯ > π π¦ , { π¦ } , ; { π₯ β } = min { X } = {οΈ { π₯ } , π π₯ (cid:54) π π¦ , { π¦ } , ; (11.4) π€ β = max { Λ π€ } = {οΈ π€ π₯ , π€ π₯ > π€ π¦ ,π€ π¦ , ; π€ β = min { Λ π€ } = {οΈ π€ π₯ , π€ π₯ (cid:54) π€ π¦ ,π€ π¦ , ; (11.5) π β = max { Λ π } = {οΈ π π₯ , π π₯ > π π¦ ,π π¦ , ;= max {οΈ min { π€ π₯ , β π€ π₯ } , min { π€ π¦ , β π€ π¦ } }οΈ ,π β = min { Λ π } = {οΈ π π₯ , π π₯ (cid:54) π π¦ ,π π¦ , ;= min {οΈ min { π€ π₯ , β π€ π₯ } , min { π€ π¦ , β π€ π¦ } }οΈ . (11.6)In such invariant denotations, it is not diο¬cult towrite down the general recurrence formula for thehalf-rare 2-Kopula of the doublet X , united combin-ing both orders: K (οΈ Λ π ( π | π// X ) )οΈ = K ( π | X // X ) (οΈ Λ π ( π | π// X ) )οΈ == β§βͺβͺβͺβ¨βͺβͺβͺβ© K β² ( π π₯π¦ (Λ π )) , π = X , K β²β² (οΈ π β β π π₯π¦ (Λ π ) )οΈ , π = { π₯ β } , K β² (1 β π π₯π¦ (Λ π )) β π β , π = { π₯ β } , K β²β² (οΈ β π β + π π₯π¦ (Λ π ) )οΈ β π β , π = β (11.7)where by Deο¬nition (11.3) π β = max {οΈ min { π€ π₯ , β π€ π₯ } , min { π€ π¦ , β π€ π¦ } }οΈ ,π β = min {οΈ min { π€ π₯ , β π€ π₯ } , min { π€ π¦ , β π€ π¦ } }οΈ .
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The mutual set-phenomenon inversion of 2-Kopulas of half-rare Λ π and free Λ π€ marginal proba-bilities has the form: K (οΈ Λ π ( π | π// X ) )οΈ = K ( π | π// X ) ( Λ π€ ) , K ( π | π// X ) (Λ π ) = K (οΈ Λ π€ ( π | π// X ) )οΈ ; (11.8) K ( π | π// X ) (οΈ Λ π ( π | π// X ) )οΈ = K ( π | X // X ) (οΈ Λ π ( π | ( π π Ξ π ) π // X ) )οΈ = K (οΈ Λ π ( π | ( π π Ξ π ) π // X ) )οΈ . (11.9)For example, for π β X = { π₯, π¦ } K ( π | X // X ) (οΈ Λ π ( π | π// X ) )οΈ = K (οΈ Λ π ( π | π// X ) )οΈ , K ( π |{ π₯ } // X ) (οΈ Λ π ( π | π// X ) )οΈ = K (οΈ Λ π ( π | ( { π¦ } Ξ π ) π // X ) )οΈ , K ( π |{ π¦ } // X ) (οΈ Λ π ( π | π// X ) )οΈ = K (οΈ Λ π ( π | ( { π₯ } Ξ π ) π // X ) )οΈ , K ( π |β // X ) (οΈ Λ π ( π | π// X ) )οΈ = K (οΈ Λ π ( π | π π // X ) )οΈ . (11.10) Λ π -ordering ahalf-rare s.e. Λ π -ordering ahalf-rare doublets of events Let us explain the role of Λ π -ordering in the framemethod using the example of constructing a 2-Kopula of the Λ π -ordered half-rare events X = { π₯, π¦ } with X -set of marginal probabilities of events Λ π = { π π₯ , π π¦ } , that is, / (cid:62) π π₯ (cid:62) π π¦ : K β² (οΈ Λ π ( π | π// { π₯,π¦ } ) )οΈ = β§βͺβͺβͺβ¨βͺβͺβͺβ© π π₯π¦ , π = { π₯, π¦ } ,π π₯ β π π₯π¦ , π = { π₯ } ,π π¦ β π π₯π¦ , π = { π¦ } , β π π₯ β π π¦ + π π₯π¦ , π = β , (11.11)where, when selected as a function parameter π π₯π¦ of the 1-Kopulas of inserted half-rare monoplates π³ β² = { π β² } = { π₯ β© π¦ } and π³ β²β² = { π β²β² } = { π₯ π β© π¦ } , areequal, respectively: K β² (οΈ Λ π ( π | π// π³ β² ) )οΈ = β§βͺβͺβͺβ¨βͺβͺβͺβ© π π β² π‘ β² , π = { π β² , π‘ β² } ,π π β² β π π β² π‘ β² , π = { π β² } ,π π‘ β² β π π β² π‘ β² , π = { π‘ β² } , β π π β² β π π‘ β² + π π β² π‘ β² , π = β , K β²β² (οΈ Λ π ( π | π// π³ β²β² ) )οΈ == β§βͺβͺβͺβ¨βͺβͺβͺβ© π π β²β² π‘ β²β² , π = { π β²β² , π‘ β²β² } ,π π¦ β π π β² β π π β²β² π‘ β²β² , π = { π β²β² } ,π π§ β π π‘ β² β π π β²β² π‘ β²β² , π = { π‘ β²β² } , β π π¦ β π π§ + π π β² + π π‘ β² + π π β²β² π‘ β²β² , π = β , (11.12) under the assumption that inserted half-raremonoplets have βequally directβ Λ π -orders: π π¦ (cid:62) π π β² (cid:62) π π‘ β² ,π π§ (cid:62) π π β²β² (cid:62) π π‘ β²β² . (11.13)However, nothing prevents the emergence of twomore βopposite Λ π ordersβ on the inserted half-raremonoplets: π π¦ (cid:62) π π‘ β² (cid:62) π π β² ,π π§ (cid:62) π π β²β² (cid:62) π π‘ β²β² . (11.14) π π¦ (cid:62) π π β² (cid:62) π π‘ β² ,π π§ (cid:62) π π‘ β²β² (cid:62) π π β²β² ; (11.15)except for the βequally inverseβ Λ π -order π π¦ (cid:62) π π‘ β² (cid:62) π π β² ,π π§ (cid:62) π π‘ β²β² (cid:62) π π β²β² , (11.16)which can not be due to the consistency of the func-tional parameters, i.e., because π π¦ = π π β² + π π β²β² (cid:62) π π‘ β² + π π‘ β²β² = π π§ . (11.17) Λ π -ordering thehalf-rare triplets of events Let us explain the role of Λ π -ordering in the framemethod using the example of constructing a 3-Kopula of the Λ π -ordered half-rare events X = { π₯, π¦, π§ } with X -set of marginal probabilities ofevents Λ π = { π π₯ , π π¦ , π π§ } , that is, / (cid:62) π π₯ (cid:62) π π¦ (cid:62) π π§ : K (οΈ Λ π ( π | π// { π₯,π¦,π§ } ) )οΈ == β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© K β² ( π π β² , π π‘ β² ) , π = { π₯, π¦, π§ } , K β² ( π π β² , β π π‘ β² ) , π = { π₯, π¦ } , K β² (1 β π π β² , π π‘ β² ) , π = { π₯, π§ } , K β² (1 β π π β² , β π π‘ β² ) β π π₯ , π = { π₯ } , K β²β² ( π π¦ β π π β² , π π§ β π π‘ β² ) , π = { π¦, π§ } , K β²β² ( π π¦ β π π β² , β π π§ + π π‘ β² ) , π = { π¦ } , K β²β² (1 β π π¦ + π π β² , π π§ β π π‘ β² ) , π = { π§ } , K β²β² (1 β π π¦ + π π β² , β π π§ + π π‘ β² ) β π π₯ , π = β , (11.18)where, when selected as function parameters π π β² , π π‘ β² , π π β² π‘ β² and π π β²β² π‘ β²β² and despite the fact that π π β²β² = π π¦ β π π β² , π π‘ β²β² = π π§ β π π‘ β² , the 2-Kopulas of insertedhalf-rare doublets π³ β² = { π β² , π‘ β² } = { π₯ β© π¦, π₯ β© π§ } and OROBYEV π³ β²β² = { π β²β² , π‘ β²β² } = { π₯ π β© π¦, π§ π β© π§ } are equal, respectively: K β² (οΈ Λ π ( π | π// π³ β² ) )οΈ = β§βͺβͺβͺβ¨βͺβͺβͺβ© π π β² π‘ β² , π = { π β² , π‘ β² } ,π π β² β π π β² π‘ β² , π = { π β² } ,π π‘ β² β π π β² π‘ β² , π = { π‘ β² } , β π π β² β π π‘ β² + π π β² π‘ β² , π = β , K β²β² (οΈ Λ π ( π | π// π³ β²β² ) )οΈ == β§βͺβͺβͺβ¨βͺβͺβͺβ© π π β²β² π‘ β²β² , π = { π β²β² , π‘ β²β² } ,π π¦ β π π β² β π π β²β² π‘ β²β² , π = { π β²β² } ,π π§ β π π‘ β² β π π β²β² π‘ β²β² , π = { π‘ β²β² } , β π π¦ β π π§ + π π β² + π π‘ β² + π π β²β² π‘ β²β² , π = β , (11.19)under the assumption that the inserted half-raredoublets have βequally directβ Λ π -orders: π π¦ (cid:62) π π β² (cid:62) π π‘ β² ,π π§ (cid:62) π π β²β² (cid:62) π π‘ β²β² . (11.20)However, nothing prevents the emergence of twomore βopposite Λ π ordersβ on the inserted half-raredoublets: π π¦ (cid:62) π π‘ β² (cid:62) π π β² ,π π§ (cid:62) π π β²β² (cid:62) π π‘ β²β² . (11.21) π π¦ (cid:62) π π β² (cid:62) π π‘ β² ,π π§ (cid:62) π π‘ β²β² (cid:62) π π β²β² ; (11.22)except for the βequally inverseβ Λ π -order π π¦ (cid:62) π π‘ β² (cid:62) π π β² ,π π§ (cid:62) π π‘ β²β² (cid:62) π π β²β² , (11.23)which can not be due to the consistency of the func-tional parameters, i.e., because π π¦ = π π β² + π π β²β² (cid:62) π π‘ β² + π π‘ β²β² = π π§ . (11.24) Λ π -non-ordered half-rare s.e.βs Above we outlined the frame method for construct-ing π -Kopulas of Λ π - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered half-rare π -s.e.βs. Itremains to extend it to construct π -Kopulas of Λ π - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) disordered half-rare π -s.e.βs using the followingtechnique, based on the obvious invariance prop-erty of permutations of events in s.e.: βas eventsfrom some s.e. do not order, the s.e. will not changeβ; and very useful in practical calculations.We denote by X * = { π₯ * , π₯ * , ..., π₯ * π β } , (11.25)β the Λ π - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered half-rare π -s.e. , which consistsfrom the same events, that an βarbitraryβ Λ π - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) non-ordered half-rare π -s.e. X = { π₯ , π₯ , ..., π₯ π β } , (11.26) i.e., X * = { π₯ * , π₯ * , ..., π₯ * π β } = { π₯ , π₯ , ..., π₯ π β } = X , (11.27)but arranged in descending order of their probabil-ities. In other words, the X * -set of marginal proba-bilities Λ π * = { π * , π * , ..., π * π β } , (11.28)is such that / (cid:62) π * (cid:62) π * (cid:62) ... (cid:62) π * π β (11.29)where π * = max { π π₯ : π₯ β X } ,π * = max { π π₯ : π₯ β X β { π₯ * }} ,...π * π +1 = max { π π₯ : π₯ β X β { π₯ * , ..., π₯ * π }} ,...π * π = max { π π₯ : π₯ β X β { π₯ * , ..., π₯ * π β }} . (11.30)Consequently, X * -set of marginal probabilities Λ π * which consists of the same probabilities that X -setof marginal probabilities Λ π , i.e., Λ π * = { π * , π * , ..., π * π β } = { π , π , ..., π π β } = Λ π, (11.31)but arranged in descending order.Now, to construct the π -Kopulas of the Λ π - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) disordered π -s.e X by the frame method it is suf-ο¬cient to construct this π -Kopula of the Λ π - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered π -s.e X * = X by this method, reasoning by (11.27)and (11.31) reasoning that π - Λ π - π -s.e. X π - Λ π - π -s.e. X * = X , reasoningby virtue of (11.27) and (11.31), that K (Λ π ) = K (Λ π * ) , (11.32)i.e., for π β X K (οΈ Λ π ( π | π// X ) )οΈ = K (οΈ Λ π * ( π | π * // X * ) )οΈ (11.33)where π * = { π₯ * : π₯ β π } β X * (11.34)are subsets of the Λ π - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered ( π β -s.e. X * . Note 21 (properties of functions of an unordered setof arguments).
Equations (11.32) and (11.33) shouldnot be regarded as a unique property of the Kopulainvariance with respect to permutations of itsarguments. This property is possessed by anyKopula, since it is a function of an unordered setof arguments. Therefore it is quite natural thatthe Kopula is invariant under permutations ofthe arguments, like any other such function. Thisproperty must be remembered only in practical
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ONFERENCE calculations, when we volence-nolens must in-troduce an arbitrary order on a disordered set inorder to be able to perform calculations.Consider the examples of Kopulas of arbitrary, i.e., Λ π -disordered, s.e.βs X = { π₯ , π₯ , ..., π₯ π β } in the notation just introduced, assuming that wehave available Kopulas of the Λ π -ordered π -s.eβs X * = { π₯ * , π₯ * , ..., π₯ * π β } = X for π = 1 , . Example 2 (invariant formula for the 2-Kopula of ahalf-rare doublet of events).
Let X = { π₯ , π₯ } bethe Λ π -non-ordered half-rare doublet of events.Then its 2-Kopula is calculated at each point Λ π ( π | π// { π₯ ,π₯ } ) β [0 , β X by the following formulas: K (οΈ Λ π ( π | π// { π₯ ,π₯ } ) )οΈ = K (οΈ Λ π ( π | π * // { π₯ * ,π₯ * } ) )οΈ == β§βͺβͺβͺβ¨βͺβͺβͺβ© K β² ( π π β² ) , π * = { π₯ * , π₯ * } , K β²β² ( π β π π β² ) , π * = { π₯ * } , K β² (1 β π π β² ) β π , π * = { π₯ * } , K β²β² (1 β π + π π β² ) β π , π * = β , (11.35)where π³ β² = { π β² } = { π₯ * β© π₯ * } , π³ β²β² = { π β²β² } = { ( π₯ * ) π β© π₯ * } . (11.36) Example 3 (invariant formula for the 3-Kopula of ahalf-rare triplet of events).
Let X = { π₯ , π₯ , π₯ } be the Λ π -non-ordered half-rare triplet of events.Then its 3-Kopula is calculated at each point Λ π ( π | π// { π₯ ,π₯ ,π₯ } ) β [0 , β X by the following formulas: K (οΈ Λ π ( π | π// { π₯ ,π₯ ,π₯ } ) )οΈ = K (οΈ Λ π ( π | π * // { π₯ * ,π₯ * ,π₯ * } ) )οΈ == β§βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβͺβ© K β² ( π π β² , π π‘ β² ) , π * = { π₯ * , π₯ * , π₯ * } , K β² ( π π β² , β π π‘ β² ) , π * = { π₯ * , π₯ * } , K β² (1 β π π β² , π π‘ β² ) , π * = { π₯ * , π₯ * } , K β² (1 β π π β² , β π π‘ β² ) β π , π * = { π₯ * } , K β²β² ( π β π π β² , π β π π‘ β² ) , π * = { π₯ * , π₯ * } , K β²β² (1 β π + π π β² , π β π π‘ β² ) , π * = { π₯ * } , K β²β² ( π β π π β² , β π + π π‘ β² ) , π * = { π₯ * } , K β²β² (1 β π + π π β² , β π + π π‘ β² ) β π , π * = β , (11.37)where when selecting as function parameters π π β² , π π‘ β² , π π β² π‘ β² π π β²β² π‘ β²β² and despite the fact that π π β²β² = π β π π β² , π π‘ β²β² = π β π π‘ β² , 2-Kopulas of the inserted half-rare doublets π³ β² = { π β² , π‘ β² } = { π₯ * β© π₯ * , π₯ * β© π₯ * } and π³ β²β² = { π β²β² , π‘ β²β² } = { ( π₯ * ) π β© π₯ * , ( π₯ * ) π β© π₯ * } are equal re- spectively: K β² (οΈ Λ π ( π | π// π³ β² ) )οΈ = β§βͺβͺβͺβ¨βͺβͺβͺβ© π π β² π‘ β² , π = { π β² , π‘ β² } ,π π β² β π π β² π‘ β² , π = { π β² } ,π π‘ β² β π π β² π‘ β² , π = { π‘ β² } , β π π β² β π π‘ β² + π π β² π‘ β² , π = β , K β²β² (οΈ Λ π ( π | π// π³ β²β² ) )οΈ == β§βͺβͺβͺβ¨βͺβͺβͺβ© π π β²β² π‘ β²β² , π = { π β²β² , π‘ β²β² } ,π β π π β² β π π β²β² π‘ β²β² , π = { π β²β² } ,π β π π‘ β² β π π β²β² π‘ β²β² , π = { π‘ β²β² } , β π β π + π π β² + π π‘ β² + π π β²β² π‘ β²β² , π = β . (11.38) For a subset of events π β X π -phenomenonrenumbering of the terrace events, generated by ( π β -s.h-r.e. X , is based on the replacement ofevents from the subset π π = X β π by their comple-ments: X ( π | π ) = π + ( π π ) ( π ) == { π₯, π₯ β π } + { π₯ π , π₯ β π π } , (11.39)from which the mutually inverse set-phenomenonrenumbering formulas follow: ter (οΈ π ( π | π β© π ) // X ( π | π ) )οΈ = ter (( π β π ) π // X ) , ter ( π// X ) = ter (οΈ (( π β π ) π ) ( π | π β© ( π Ξ π ) π ) // X ( π | π ) )οΈ (11.40)for π β X and π β X .Therefore, the π -phenomenon renumbering of theterrace events, generated by ( π β -s.h-r.e. X , by theformulas (11.40) is geometrically interpreted on the ( π β -dimensional Venn diagram of this s.e. as areο¬ection of the X -hypercube relative to those hy-perplanes that are orthogonal to the π₯ -axes num-bered by the events π₯ β π π β X (see Fig. 21 for thedoublet of events). X -simplex on the X -hypercube Take an arbitrary ( π β -s.e. X β π with e.p.d. ofthe 1st kind, which, as is known [1], is deο¬ned asthe X -set of probabilities of terrace events of the1st kind { π ( π// X ) , π β X } . (11.41)Look at the X -set (11.41) as a X -hyper-point from ahalf-rare π -vertex simplex S X == β§β¨β© { π ( π// X ) , π β X } : π ( π// X ) (cid:62) , βοΈ π β X π ( π// X )=1 β«β¬β , (11.42) OROBYEV { π₯, π¦ π } { π¦ π } { π¦ π } { π₯ π , π¦ π } X ( π | π₯ ) X ( π |β ) { π₯ } β β { π₯ π }{ π₯ } β β { π₯ π } X ( π | π₯π¦ ) X ( π | π¦ ) { π₯, π¦ } { π¦ } { π¦ } { π₯ π , π¦ } Figure 21: Geometric interpretation of a set-phenomenon renumberingthe terraced events, generated by the doublet of half-rare events X = { π₯, π¦ } = { π₯, π¦ } ( π | π₯π¦ ) , by reο¬ections with respect to straight lines or-thogonal to the coordinate axes and intersecting them at the points 1/2.In the form of unit squares (with the origin in the bottom left cornerof each square), the four Venn diagrams of doublets of half-rare events X = X ( π | π₯π¦ ) and its set-phenomena X ( π | π₯ ) = { π₯, π¦ π } , X ( π | π¦ ) = { π₯ π , π¦ } and X ( π |β ) = { π₯ π , π¦ π } are shown; the terrace events are marked withsubsets of the doublet of half-rare events X or its set-phenomena, consist-ing of both half-rare events and its complements; on each diagram pairsof events, from which the doublet of half-rare events or its set-phenomenaconsists, are shaded (aqua). to each vertex of which the degenerate e.p.d. cor-responds. In this e.p.d., as is known, only one ofthe 1st kind of probability, equal to one, is differentfrom zero. Number the vertex X of the simplex S X by the subset π β X . The degenerate e.p.d. of the1st kind with π ( π// X ) = 1 corresponds to this ver-tex. And associate the vertex π ππππ with the hy-percube [0 , ππ‘ππππ πππππ , numbered by the π ππππ -set:, e.p.d. , , , of the 1st kind, , . π β X X - S X , e.p.d. of the 1st kind π ( π// X ) = 1 . And associatewith it the vertex of X -hypercube [0 , β X , number-ing by the following X -set: { Ξ₯ π// X ( π₯ ) , π₯ β X } (11.43)where Ξ₯ π// X ( π₯ ) = {οΈ , π₯ β π, , , (11.44)are values of the indicator of subset π β X onevents π₯ β X . Deο¬ne a prohection pr : S X β [0 , β X of the X -simplex S X on X -hypercube [0 , β X by thefollowing formula: pr ( { π ( π// X ) , π β X } ) = { π π₯ , π₯ β X } (11.45) where π π₯ = βοΈ π β X π ( π// X )Ξ₯ π// X ( π₯ ) == βοΈ π₯ β π β X π ( π// X ) (11.46)is a convex combination of hypercube vertices,which, as known [1], is intepreted as the probabil-ity of event π₯ β X .With projection (11.45) vertices of the X -simplexmaps to vertices of the X -hypercube, and edgesmap to its edges or diagonals (see [9], [10] and Fig.23). Example 4 (projections of vertices of a X -simplex). For example, the vertex of the X -simplex enu-merated by the subset π β X corresponds to thedegenerate e.p.d. of the 1st kind with probabilities π ( π// X ) = {οΈ , π = π , , π ΜΈ = π β X . From (11.46) you obtain that π π₯ = Ξ₯ π // X ( π₯ ) = {οΈ , π₯ β π , , π₯ β X β π . Therefore, by (11.43) { π π₯ , π₯ β X } = { Ξ₯ π // X ( π₯ ) , π₯ β X } is a vertex of the X -hypercube.In particular, the β -vertex of the X -simplex, i.e., thevertex numbered by the subset π = β is projectedinto the X -set { , ..., } , consisting of the zero prob-abilities of marginal events, in other words, pro-jected into the β -vertex of the X -hypercube, i.e., tothe vertex located at the beginning coordinates: { , ..., } βΌ π ( π// X ) = {οΈ , π = β , , β ΜΈ = π β X ; (11.47)and the X -vertex of the X -simplex, i.e., the vertexenumerated by the subset π = X is projected intothe X -set { , ..., } , consisting of the unit probabil-ities of marginal events, in other words, projectedinto the X -vertex of the X -hypercube, i.e., to the ver-tex opposite to the origin: { , ..., } βΌ π ( π// X ) = {οΈ , π = X , , X ΜΈ = π β X . (11.48)In general, due to the linearity of the projection(11.45), the set of such points of the X -simplex thatproject into the same point of the X -hypercube isconvex and forms a sub-simplex of smaller dimen-sion.
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ONFERENCE ( π β -diagram We will ο¬gure out how a Venn ( π β -diagram ofan arbitrary ( π β -s.e. is constructed on the ba-sis of the projection (11.45), in which the role of thespace of universal elementary events β¦ is playedby the unit ( π β βdimensional hypercube . Such aVenn ( π β -diagram puts terraced hypercubes gen-erated by dividing a unit hypercube in half orthog-onal to each of the π axes into a one-to-one corre-spondence with the terraced events generated bythe given ( π β -s.e.Take ο¬rst ( π β -s.h-r.e. X and represent its Venn ( π β -diagram On which β¦ is represented by a unit ( π β -dimensional hypercube that serves as or-dered image of the X -hypercube [0 , β X = β¨οΈ π₯ β X [0 , π₯ , (11.49)broken by hyperplanes orthogonal to π₯ -axis and in-tersecting them at points / into π π -terraced hy-percubes for π β X [0 , β ter ( π// X ) = β¨οΈ π₯ β π [0 , / π₯ β¨οΈ π₯ β X β π (1 / , π₯ , (11.50)where each marginal half-rare event π₯ β X is rep-resented as a π₯ -half of X -hypercube containing theorigin : [0 , / π₯ β [0 , β ( X β{ π₯ } ) , (11.51)its complement π₯ π = β¦ β π₯ is represented in theform of another π₯ -half of X -hypercube that does notcontain the origin : (1 / , π₯ β [0 , β ( X β{ π₯ } ) , (11.52)and the π -terraced event ter ( π// X ) β as a π -terraced hypercube (11.50): ter ( π// X ) βΌ [0 , β ter ( π// X ) . (11.53)The formula (11.53) once again points to a one-to-one correspondence between the π -space of ter-raced hypercubes (11.50) from the Venn ( π β -diagram of ( π β -s.h-r.e. X and π -totality of ter-raced events, generated by X .If the correspondence between the terraced hyper-cubes and the terraced events looks natural, thenfor the sets of half-rare events X the correspon-dence between the terraced hypercubes and thenumbering of the vertices of the X -simplex pro-jected into the corresponding vertices of the X -hypercube under the projection (11.45) is deο¬nedby the operation of the complement and requires aspecial see the Venn 2-diagram doublet of half-rare events in Fig. 23. The role of the order of events in s.e. when working withtheir images in R π is discussed in [ ? ]. Note 22 (correspondence between the numbering of ter-raced hypercubes and vertices of X -simplex of e.p.d.βsof the 1st kind of ( π β -s.h-r.e. X on its Venn ( π β -diagram). On the Venn ( π β -diagram of ( π β -setof half-rare events X , every π π -terraced hyper-cube [0 , β ter ( π π // X ) contains the π -vertex of X -hypercube, into which corresponding π -vertexof X -simplex S X of e.p.d.βs of the 1st kind of ( π β -s.h-r.e. X for π β X is projected: { , π₯ β π } + { , π₯ β X β π } β [0 , β ter ( π π // X ) , (11.54)in particular, for β -vertex and X -vertex we have: { , π₯ β X } = { , ..., } β [0 , β ter ( X // X ) , { , π₯ β X } = { , ..., } β [0 , β ter ( β // X ) . (11.55), ( π β - - ..Β΄. ( π β - X events [6].It is not diο¬cult to guess that the Venn ( π β -diagram of an arbitrary set-phenomenon of s.h-r.e.differs from the Venn ( π β -diagram of X itselfonly by renumbering terraced events using formu-las from [6]. Letβs do Note 23 (Venn ( π β -diagram of set-phenomena of a setof half-rare events). On the Venn ( π β -diagramof the π -phenomenon X ( π | π ) of ( π β -set of half-rare events X , every π β π -terraced hyper-cube [0 , β ter ( π Ξ π// X ) contains the π -vertex of X -hypercube, into which the corresponding π -vertexof X -simplex S X of e.p.d. of the 1st kind of ( π β -s.h-r.e. X for π β X and π β X is projected: { , π₯ β π } + { , π₯ β X β π } β [0 , β ter ( π Ξ π// X ) , (11.56)in particular, for the β -phenomenon X ( π |β ) = X ( π ) and for β -vertex and X -vertex we have: { , π₯ β X } = { , ..., } β [0 , β ter ( β // X ( π ) ) , { , π₯ β X } = { , ..., } β [0 , β ter ( X ( π ) // X ( π ) ) . (11.57) X -hypercube Definition 11 ( set-phenomenon spectrum of func-tions on the X -hypercube ). With each function π β Ξ¨ X , deο¬ned on the X -hypercube, the π func-tions are connected. These functions are deο¬nedon the X -hypercube by formulas: π π ( Λ π€ ) = π (οΈ Λ π€ ( π | π// X ) )οΈ for π β X . The family of the all such functions { π π : π β X } is called the set-phenomenon X -spectrum of thefunction π . OROBYEV { π¦ } { π₯, π¦ } π π₯ (cid:62) β π π¦ β π π₯ (cid:62) β π π¦ β π π¦ > π π₯ β π π¦ > β π π₯ π π¦ > π π₯ π π¦ > β π π₯ π π₯ (cid:62) π π¦ β π π₯ (cid:62) π π¦ β { π₯ } ter ( π₯//π₯π¦ ) ter ( β //π₯π¦ ) ter ( π¦//π¦π₯ ) ter ( β //π¦π₯ ) ter ( π¦π₯//π¦π₯ ) ter ( π₯//π¦π₯ ) ter ( π₯π¦//π₯π¦ ) ter ( π¦//π₯π¦ ) Figure 22: The projection of a simplex (tetrahedron) of doublets events S { π₯,π¦ } on a unit { π₯, π¦ } -square [0 , β{ π₯,π¦ } of its marginal probabilities Λ π = { π π₯ , π π¦ } . The π - of this simplex are projected in corresponding π -vertices of { π₯, π¦ } -square, π β { π₯, π¦ } , and the all of e.p.d.βs of dou-blets of events with given { π₯, π¦ } -set of probabilities of marginal events Λ π = { π π₯ , π π¦ } are projected in each point Λ π β [0 , β{ π₯,π¦ } . In the leftdown quadrant (aqua) e.p.d.βs of the all of doublets of half-rare events oftwo kind are projected: / (cid:62) π π₯ (cid:62) π π¦ (unshaded) and π π₯ < π π¦ (cid:54) / (shaded triangle); in the remaining 3 quadrants e.p.d.βs of β -phenomena, { π¦ } -phenomena and { π₯ } -phenomena of doublets of half-rare events areprojected. The half-rare doublets of the second kind: π π₯ < π π¦ (cid:54) / ,are projected in the shaded triangle of left down quadrant, and its set-phenomena β in shaded triangles of corresponding quadrants. The ter-raced events, generated by doublets of half-rare events { π₯, π¦ } , are markedby the white formulas. Letβs deο¬ne for each π β X the terraced π -hypercube ter β ( π// X ) = β¨οΈ π₯ β π [οΈ , ]οΈ β¨οΈ π₯ β X β π [οΈ , )οΈ , from which the X -hypercube is composed: [0 , β X = βοΈ π β X ter β ( π// X ) . Lemma 6 ( on a set-phenomenon X -spectrum of nor-malized function ). In order that the family offunctions { π π : π β X } Ξ¨ X is a set-phenomenon X -spectrum of some function normalized on the X -hypercube, it is necessary and suο¬cient that βοΈ π β X π π ( Λ π€ ) = 1 (11.58)for all Λ π€ β [0 , β X . Proof.
1) If the family { π π : π β X } is aset-phenomenon X -spectrum of some normalizedfunction, then by Deο¬nition 11 the equality (11.58)is satisο¬ed. 2) Let now the equality (11.58) is satis-ο¬ed. Construct the function π on the X -hypercube { π¦ } { π₯, π¦ } π π₯ (cid:62) β π π¦ β π π₯ (cid:62) β π π¦ β π π¦ > π π₯ β π π¦ > β π π₯ π π¦ > π π₯ π π¦ > β π π₯ π π₯ (cid:62) π π¦ β π π₯ (cid:62) π π¦ β { π₯ } { π₯ } β { π¦ } β { π¦, π₯ } { π₯ }{ π₯, π¦ } { π¦ } Figure 23: The projection of a simplex (tetrahedron) of doublets events S { π₯,π¦ } on a unit { π₯, π¦ } -square [0 , β{ π₯,π¦ } of its marginal probabilities Λ π = { π π₯ , π π¦ } . The π - of this simplex are projected in corresponding π -vertices of { π₯, π¦ } -square, π β { π₯, π¦ } , and the all of e.p.d.βs of dou-blets of events with given { π₯, π¦ } -set of probabilities of marginal events Λ π = { π π₯ , π π¦ } are projected in each point Λ π β [0 , β{ π₯,π¦ } . In the leftdown quadrant (aqua) e.p.d.βs of the all of doublets of half-rare events oftwo kind are projected: / (cid:62) π π₯ (cid:62) π π¦ (unshaded) and π π₯ < π π¦ (cid:54) / (shaded triangle); in the remaining 3 quadrants e.p.d.βs of β -phenomena, { π¦ } -phenomena and { π₯ } -phenomena of doublets of half-rare events areprojected. The half-rare doublets of the second kind: π π₯ < π π¦ (cid:54) / ,are projected in the shaded triangle of left down quadrant, and its set-phenomena β in shaded triangles of corresponding quadrants. The ter-raced events, generated by doublets of half-rare events { π₯, π¦ } , are markedby the white formulas. by the following way π ( Λ π€ ) = β§βͺβͺβͺβͺβ¨βͺβͺβͺβͺβ© π β (οΈ Λ π€ ( π |β ) )οΈ , Λ π€ β [0 , / β X ,. . . ,π π (οΈ Λ π€ ( π | π// X ) )οΈ , Λ π€ β ter β ( π// X ) ,. . . ,π X (οΈ Λ π€ ( π | X ) )οΈ , Λ π€ β [1 / , β X and show that the π is normalized on the X -hypercube. Indeed, noting that for an arbitrary π β X the equality π ( Λ π€ ) = π π (οΈ Λ π€ ( π | π// X ) )οΈ is equiva-lent to the equality π (οΈ Λ π€ ( π | π// X ) )οΈ = π π ( Λ π€ ) , we obtainthe required: βοΈ π β X π (οΈ Λ π€ ( π | π// X ) )οΈ = βοΈ π β X π π ( Λ π€ ) = 1 . Lemma 7 ( on a set-phenomenon X -spectrum of the1-function ). In order that the family of functions { π π : π β X } Ξ¨ X is a set-pehomenon X -spectrum ofsome 1-function on the X -hypercube, it is necessaryand suο¬cient that for each π₯ β X βοΈ π₯ β π β X π π ( Λ π€ ) = π€ π₯ (11.59)
20 T HE XIV FAMEMSβ2015 C
ONFERENCE for all Λ π€ β [0 , β X . Proof.
1) If the family (1.3) is a set-phenomenon X -spectrum of some 1-function, then partial sumsof functions from the family at π₯ β π β X π€ π₯ : βοΈ π₯ β π β X π π ( Λ π€ ) = π€ π₯ for each Λ π€ β [0 , β X by Deο¬nition 4. 2) Let nowthe equalities (1.4) are satisο¬ed. Letβs construct thefunction π on X -hypercube by the following way π ( Λ π€ ) = β§βͺβͺβͺβͺβ¨βͺβͺβͺβͺβ© π β (οΈ Λ π€ ( π |β ) )οΈ , Λ π€ β [0 , / β X ,. . . ,π π (οΈ Λ π€ ( π | π// X ) )οΈ , Λ π€ β ter β ( π// X ) ,. . . ,π X (οΈ Λ π€ ( π | X ) )οΈ , Λ π€ β [1 / , β X and show that π is a 1-function on the X -hypercube.Indeed, noting that for an arbitrary π β X theequality π ( Λ π€ ) = π π (οΈ Λ π€ ( π | π// X ) )οΈ is equivalent to theequality π (οΈ Λ π€ ( π | π// X ) )οΈ = π π ( Λ π€ ) , we obtain the re-quired: βοΈ π₯ β π β X π (οΈ Λ π€ ( π | π// X ) )οΈ = βοΈ π₯ β π β X π π ( Λ π€ ) = π€ π₯ . { , } { , } π₯π¦ π₯π¦
10 00 π¦π₯ π¦π₯
01 00 π¦π₯ π¦π₯
11 10 π₯π¦ π₯π¦
11 01 { π π₯ , π π¦ } = { , } { , } { π₯ } β { π₯,π¦ } { π¦ } Figure 24: The projection of -vertices simplex on a square, on which thescheme is superimposed, illustrating a connection of two permutations ofevents in a half-rare doublet with the set-phenomena.
12 Remaining behind the scenes
In the text and, in particular, in the Appendix,the value of the Λ π -ordering condition of the set of { , , } { , , } π₯π¦π§ π₯π¦π§
100 000 π₯π§π¦ π₯π§π¦
100 000 π¦π₯π§ π¦π₯π§
010 000 π§π₯π¦ π§π₯π¦
010 000 π¦π§π₯ π¦π§π₯
001 000 π§π¦π₯ π§π¦π₯
001 000 π§π¦π₯ π§π¦π₯
011 010 π¦π§π₯ π¦π§π₯
101 100 π§π₯π¦ π§π₯π¦
011 001 π¦π₯π§ π¦π₯π§
110 100 π₯π§π¦ π₯π§π¦
101 001 π₯π¦π§ π₯π¦π§
110 010 { π π₯ , π π¦ , π π§ } = { , , } { , , } { π₯ } β { π₯,π¦ } { π¦ } { , , } { , , } π₯π¦π§ π₯π¦π§
101 001 π₯π§π¦ π₯π§π¦
110 010 π¦π₯π§ π¦π₯π§
011 001 π§π₯π¦ π§π₯π¦
110 100 π¦π§π₯ π¦π§π₯
011 010 π§π¦π₯ π§π¦π₯
101 100 π§π¦π₯ π§π¦π₯
111 110 π¦π§π₯ π¦π§π₯
111 110 π§π₯π¦ π§π₯π¦
111 101 π¦π₯π§ π¦π₯π§
111 101 π₯π§π¦ π₯π§π¦
111 011 π₯π¦π§ π₯π¦π§
111 011 { π π₯ , π π¦ , π π§ } = { , , } { , , } { π₯,π§ } { π§ }{ π₯,π¦,π§ } { π¦,π§ } Figure 25: These are not geometrical projections of -vertices simplex ona cube, but two conditional schemes of these projections, which illustratea connection of six permutations of events in a half-rare triplet of eventswith its set-phenomena. The conditional scheme of the projection onthe upper half of the cube is shown at the top, on the lower half β at thebottom. In the Venn diagram of half-rare events: π₯ is the left, π¦ is theright, and π§ is the lower half of the cube. events is speciο¬ed, which complicates the compu-tational implementation of the above algorithms inthe frame method of constructing Kopulas as setfunctions of the set of marginal probabilities. Thereason for this complication lies in the properties ofthe set-functions, i.e., functions of a set that differfrom the properties of arbitrary functions of sev-eral variables. The point is that the set-function OROBYEV of the set of marginal probabilities is necessarilya symmetric function of the marginal probabilityvector (
Cartesian representation of Kopula , see Pro-legomenon 9), to determine which it is suο¬cientto specify its values only on those vectors whosecomponents are ordered, for example, in descend-ing order, so that the remaining values can be de-termined by the appropriate permutations of thearguments. For example, the
Cartesian representa-tion of an π -Kopula in R π is suο¬cient to deο¬ne onthe /π ! part of the unit π -hypercube so that thisrepresentation becomes deο¬nite on the whole hy-percube by continuing permutations of arguments.Although this task is purely technical, but its solu-tion opens the way for the application of the pro-posed Kopula (eventological copula) theory to theconstruction of the eventological theory of ordi-nary copulas that determine the joint distributionof a given set of marginal distributions. The au-thor encountered this when developing the pro-gram code, which calculated all the illustrationsfor the Kopula examples. The problem is solvedprogrammatically, but requires a detailed descrip-tion of this solution (see Fig. 24 and 25), which,of course, together with the eventological theory ofcopula deserves a separate publication.In conclusion, I can not resist the temptation toquote the formulation of the tenth Prolegomenonof the Kopula theory, which reveals the content ofthese my next publications. Prolegomenon 10 (Cartesian representation ofthe π -Kopula defines π classical copulas of π marginal uniform distributions on [0 , ).
13 On the inevitable development oflanguage
This ο¬rst work on the theory of the eventologicalcopula is over at the end of July 2015. It sumsup the work on the eventological theory of prob-abilities, raising the theory of Kopula to its apex.The work is written in a mathematical language, inwhich the state of the eventological theory was re-ο¬ected precisely at the time when the author unex-pectedly, but by the way, got a brilliant example oftwo statisticians from sociology and ecology, whoimmediately forced him to postpone polishing ofthe Kopula theory for almost a year in order to im-mediately immerse themselves in the destructivecreation of a new unifying eventological theory ofexperience and chance by the agonizing fusion oftwo dual theories: the eventological theory of be-lievabilities and the eventological theory of proba-bilities.Because of this, the mathematical language of thiswork is just a pretension to the eventological prob-ability theory, which does not yet know that thereis a very close twin that exists β the eventologicaltheory of believabilities. Therefore, in the termi-nology of this work, those crucial changes in the basic concepts and notations that were invented toconstruct a unifying eventological theory did notο¬nd any worthy reο¬ection. Of course, the new uni-fying theory suggests the development of the orig-inal mathematical language of dual Kopulas, oneof which hosts the eventological probability theory,and the other β in the eventological believabilitytheory. (cid:70)
The English version of this article was publishedon November 12, 2017. Therefore, my later works[11, 12, 13, 14], which expand the themes of thiswork, are added to the list of references. Due tothe arXiv.org limitation on the volume of publi-cation, the work is reduced by removing some il-lustrations. In full, the work is available at: . References [1] O. Yu. Vorobyev.
Eventology . Siberian Federal Univer-sity, Krasnoyarsk, Russia, 435p., ,2007. [78, 79, 94, 107, 116, 117][2] A. Sklar. Fonctions de rΒ΄epartition Β΄a π dimensions et leursmarges. Publ. Inst. Statist. Univ. Paris , 8:229β231, 1959. [78,82, 96][3] R.B. Nelsen.
An introducvtion to copulas . Springer, NewYork, 1999. [78][4] R.B. Nelsen. Properties and applications of copulas: A briefsurvey.
Proceedings of the First Brazilian Conference on Sta-tistical Modelling in Insurance and Finance (J. Dhaene, N.Kolev, and P. Morettin, eds.), Institute of Mathematics andStatistics, University of Sao Paulo , pages 10β28, 2003. [78][5] C. Alsina, M.J. Frank, and B. Schweizer.
Assocative func-tions: Triangular Norms and Copulas . World Scientiο¬c Pub-lishing Co. Pte. Ltd., Singapore, 2006. [78][6] O. Yu. Vorobyev. Ordered sets of half-rare events andits set-penomena. In.
Proc. of the XIV Intern. FAMEMSConf. on Financial and Actuarial Mathematics and Even-tology of Multivariate Statistics, Krasnoyarsk, SFU (OlegVorobyev ed.) , pages 52β59, ISBN 978β5β9903358β5β1, , 2015. [79, 80, 83, 94, 113, 118][7] O. Yu. Vorobyev and G. M. Boldyr. On a new notion of con-ditional event and its application in eventological analysis.
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Proc. of the XIII Intern. FAMEMS Conf. on Fi-nancial and Actuarial Mathematics and Eventology of Mul-tivariate Statistics, Krasnoyarsk, SFU (Oleg Vorobyev ed.) ,pages 112β121, ISBN 978β5β9903358β4β4, 2014. [81, 111][9] O. Yu. Vorobyev. FrΒ΄echet boundary eventological distri-butions and its applications. In.
Proc. of the XV Intern.EM Conf. on Eventological Mathematics. Krasnoyarsk, KSU(Oleg Vorobyev ed.) , 1:57β69, 2010. [117][10] O. Yu. Vorobyev. Eventologial generalization of FrΒ΄echetbounds for a set of events. In.
Proc. of the XIII Intern.FAMEMS Conf. on Financial and Actuarial Mathematicsand Eventology of Multivariate Statistics, Krasnoyarsk, SFU(Oleg Vorobyev ed.) , pages 122β130, ISBN 978β5β9903358β4β4, 2014. [117][11] O. Yu. Vorobyev. Elements of the kopula (eventologi-cal copula) theory. In.
Proc. of the XIV Intern. FAMEMSConf. on Financial and Actuarial Mathematics and Even-tology of Multivariate Statistics, Krasnoyarsk, SFU (OlegVorobyev ed.) , pages 78β125, ISBN 978β5β9903358β5β1, , 2015. [121]
22 T HE XIV FAMEMSβ2015 C
ONFERENCE [12] O. Yu. Vorobyev. Theory of dual co βΌ event means. In. Proc.of the XV Intern. FAMEMS Conf. on Financial and Actuar-ial Mathematics and Eventology of Multivariate Statistics &the Workshop on Hilbertβs Sixth Problem; Krasnoyarsk, SFU(Oleg Vorobyev ed.) , pages 44β93, ISBN 978β5β9903358β6β8, , 2016. [121][13] O. Yu. Vorobyev. Postulating the theory of experience andof chance as a theory of co βΌ events (co βΌ beings). In. Proc.of the XV Intern. FAMEMS Conf. on Financial and Actuar-ial Mathematics and Eventology of Multivariate Statistics &the Workshop on Hilbertβs Sixth Problem; Krasnoyarsk, SFU(Oleg Vorobyev ed.) , pages 25β43, ISBN 978β5β9903358β6β8, , 2016. [121][14] O. Yu. Vorobyev. An element-set labelling a Cartesian prod-uct by measurable binary relations which leads to postu-lates of the theory of experience and chance as a theoryof co βΌ events. In. Proc. of the XV Intern. FAMEMS Conf.on Financial and Actuarial Mathematics and Eventology ofMultivariate Statistics & the Workshop on Hilbertβs SixthProblem; Krasnoyarsk, SFU (Oleg Vorobyev ed.) , pages 9β24,ISBN 978β5β9903358β6β8,