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 Sclarโs theorem on copulas (1959). TheKopulas of doublets and triplets of events are given,as well as of some ๐ -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 โ to buildclasses of e.p.dโs of sets of events whose events hap-pen with given probabilities of marginal events . c โ Oleg Vorobyev (ed.), Proc. XIV FAMEMSโ2015, Krasnoyarsk: SFU To distinguish the quite differently de๏ฌned notion of an eventological copula from the classical concept of copula in thesense of Sklar (1959), the following radical terminology withcapital โKโ is used:
Kopula = eventological copula; ๐ -Kopula =eventological ๐ -copula. Editing the text of November 12, 2017.
We formulate the general statement of the problemof the ๐ -Kopula theory for ๐ -sets of events. If inthe classical theory the copula is the tool for select-ing some family of joint d.f.โs of a set of random vari-ables from the set of all d.f.โs with given marginald.f.โs, then in the eventological theory the Kopula isthe tool for selecting a family of e.p.d.โs of the 1stkind of the set events from the set of all e.p.d.โs withthe given probabilities of marginal events.However, unlike the classical d.f.โs the functions ofe.p.d.โs of the 1st kind of the ๐ -s.e. X are functionsthat are de๏ฌned as the sets { ๐ ( ๐// X ) , ๐ โ X } (1.1)of all its ๐ values, probabilities of the 1st kind ๐ ( ๐// X ) , on the set of all subsets of this ๐ -s.e.So, letโs 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 ๐ probabilities ofthe 1st kind (1.1), it is necessary and su๏ฌcient tospecify a family of sets from the ๐ โ 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 ๐ probability values of the1st kind (1.1) with given probabilities of marginalevents that form the X -set ห ๐ = { ๐ ๐ฅ , ๐ฅ โ X } , (1.2)it is necessary and su๏ฌcient to specify a family ofsets from the ๐ โ ๐ โ parameters, since for eachcollection there must be another ๐ constraints forevents ๐ฅ โ X : โ๏ธ ๐ฅ โ ๐ โ X ๐ ( ๐// X ) = ๐ ๐ฅ . (1.3)Therefore, โto de๏ฌne the family of functions ofe.p.d.โs of the 1st kind with given probabilities ofmarginal eventsโ means โto de๏ฌne a family of setsfrom the ๐ โ ๐ โ parametersโ as sets of functionsof these probabilities. Eventological theory shouldsolve this problem with the help of a convenienttool, the Kopula, which allows us to de๏ฌne a fam-ily of sets from the ๐ โ ๐ โ 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 ๐ -Kopula in the eventological the-ory is an instrument for de๏ฌning the family ofprobability distributions of the 1st kind of the ๐ -s.e., in the form of a family of ๐ -set of functions ofthe probabilities of their ๐ marginal events. The main results of this paper are presented ina rather rigorous mathematical manner. And al-though the de๏ฌnitions, 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 ๐ โ X are replaced by their complements,then we get a new set of events X ( ๐ | ๐ ) = ๐ + ( X โ ๐ ) ( ๐ ) , which is called the ๐ -phenomenon of s.e. X .The set of all such ๐ -phenomena for ๐ โ 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-๏ฌnes the phenomena and phenomenon-dom ofthe set ห ๐ = { ๐ ๐ฅ , ๐ฅ โ X } of the probabilities ofmarginal events ๐ฅ โ X : the ๐ -phenomenon ห ๐ ( ๐ | ๐ ) = { ๐ ๐ฅ , ๐ฅ โ ๐ } + { โ ๐ ๐ฅ , ๐ฅ โ X โ ๐ } of the set ofmarginal probabilities ห ๐ is obtained by replacingthe marginal probabilities ๐ ๐ฅ by complementarymarginal probabilities โ ๐ ๐ฅ when ๐ฅ โ X โ ๐ . 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 ( ๐ | ๐ ) , X ( ๐ | ๐ ) set-phenomena of the setof events X their probability distributions are relatedto each other set-phenomenal transformations . ห ๐ ( ๐ | ๐ ) , ห ๐ ( ๐ | ๐ ) set-phenomena of the set ofmarginal probabilities ห ๐ events from X are also in-terconnected by set-phenomenal transformations .The event ๐ฅ โ X is called half-rare [6] if the proba-bility ๐ ๐ฅ = P ( ๐ฅ ) with which it happens is not morethan half: ๐ ๐ฅ (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 di๏ฌcult to guess that for any s.e. X , in the X phenomenon-dom of the sets of events X ( ๐ | ๐ ) , ๐ โ X , and in the X -phenomenon-dom of the sets of its marginal probabilities ห ๐ ( ๐ | ๐ ) , ๐ โ 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 ๐ -s.e.โs isgenerated by ๐ functions from half-rare variablesde๏ฌned on the half-hypercube [0 , / ๐ with valuesfrom [0 , that are continued by the set-phenomenaltransformations of the half-rare variables to the cor-responding half-hypercubes, all together completely๏ฌlling 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 = { ๐ฅ } + ๐ณ with the frame half-rare event ๐ฅ โ X ,happening with the highest probability among allevents from X , are two sets of half-rare events ๐ณ โฒ = { ๐ฅ } ( โฉ ) ๐ณ and ๐ณ โฒโฒ = { ๐ฅ ๐ } ( โฉ ) ๐ณ that partitionthe set of other events ๐ณ = X โ { ๐ฅ } into two: ๐ณ = ๐ณ โฒ (+) ๐ณ โฒโฒ . The events of one of them, ๐ณ โฒ ,are contained in the frame half-rare event ๐ฅ , andthe events of the other, ๐ณ โฒโฒ , are contained in itscomplement ๐ฅ ๐ = โฆ โ ๐ฅ . Prolegomenon 6 (the insertable sets of events andconditional e.p.d.โs of a set of events with respectto the frame event and its complement).
Conditionale.p.d.โs of the 1st kind of one s.e. X with respectto the other s.e. Y are de๏ฌned in the traditionalway [1]. However, until now attempts to de๏ฌnesuch a โconditionalโ 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.โs in a frame event is a well-de๏ฌnedโersatzโ of such โconditionalโ s.e.โs. The e.p.d.โsof this โersatzโ, although they do not coincidewith two conditional e.p.d.โs 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.โs of two frame s.e.โscharacterize the corresponding two conditionede.p.d.โs 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.โs concerning theframe event and its complement. A recurrent HE XIV FAMEMSโ2015 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.โs 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.โs 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 ๐ -Kopula in R ๐ ). It follows from the Prolegomenon4 that the Cartesian representation of the ๐ -Kopulain R ๐ should be a symmetric function of ๐ orderedvariables, marginal probabilities of events from ๐ -s.e. X , which is de๏ฌned on the ๐ -dimensional unithypercube [0 , ๐ . The Cartesian representation ofthe ๐ -Kopula is based on the fact that its symmetricimage takes the same values on all permutations ofits arguments, that is, is de๏ฌned by the permutationof ๐ events group. Moreover, the value of sucha symmetric function on an arbitrary ๐ -vector ยฏ ๐ค = { ๐ค , ..., ๐ค ๐ } โ [0 , ๐ is equal to the valueof the ๐ -Kopula on an ordered X -set of marginalprobabilities of half-rare events ห ๐ = { ๐ ๐ฅ , ๐ฅ โ X } ,the ordered half-rare projection of the ๐ -vector ยฏ ๐ค ,the order of the variables in which is de๏ฌned by an ๐ -permutation ๐ ยฏ ๐ค that has the components of thehalf-rare projection ยฏ ๐ค * in decreasing order, where ๐ค * ๐ = {๏ธ ๐ค ๐ , ๐ค ๐ (cid:54) / , โ ๐ค ๐ , ๐ค ๐ > / (1.4) are components of the ๐ -vector ยฏ ๐ค * of the half-rareprojection of the ๐ -vector ยฏ ๐ค , ๐ = 1 , ..., ๐ . As a result,the ordered half-rare X is the set of marginal prob-abilities ห ๐ = ห ๐ ( ยฏ ๐ค ) , on which the ๐ -Kopula takes thesame value as a symmetric function on ยฏ ๐ค , is given bythe formula ห ๐ ( ยฏ ๐ค ) = ๐ ยฏ ๐ค ( ยฏ ๐ค * ) , (1.5) which de๏ฌnes the Cartesian representation of the ๐ -Kopula in R ๐ for each ยฏ ๐ค โ [0 , ๐ . We consider the general probability space of Kol-mogorov events (โฆ , ๐ (cid:102) , P ) , some particular proba-bility space of events (โฆ , ๐ , P ) and the ๐ - set of events( ๐ -s.e.) X โ ๐ with the event-probability distribu-tion (e.p.d. ) of the 1st kind ๐ ( X ) = { ๐ ( ๐// X ) : ๐ โ 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 ๐ X = { ๐ ๐// X : ๐ โ X } , which, recall, are related to each other by the Mo-bius inversion formulas: ๐ ๐// X = โ๏ธ ๐ โ ๐ ๐ ( ๐ // X ) ,๐ ( ๐// X ) = โ๏ธ ๐ โ ๐ ( โ | ๐ |โ| ๐ | ๐ ๐// X . Definition 1 (set-phenomena of a s.e. and itsphenomenon-dom).
Every ๐ -s.e. X โ ๐ generates itsown X -phenomenon-dom , de๏ฌned as a ๐ -family ( ๐ | X ) = {๏ธ X ( ๐ | ๐ ) , ๐ โ X }๏ธ , (2.1)composed of ๐ -s.e. in the form X ( ๐ | ๐// X ) = X ( ๐ | ๐ ) = ๐ + ( X โ ๐ ) ( ๐ ) โ ๐ , which for each ๐ โ X is called its set-phenomen [6], more precisely , ๐ -phenomen , where ๐ ( ๐ ) = { ๐ฅ ๐ : ๐ฅ โ ๐ } is an -complement of the s.e. ๐ โ X .We also recall that probabilities of the second kind ๐ ๐ฅ = ๐ { ๐ฅ } = P โโ โ๏ธ ๐ฅ โ{ ๐ฅ }โ X ๐ฅ โโ = P ( ๐ฅ ) are probabilities of marginal events from { ๐ฅ } โ X ( marginal probabilities ), probabilities of the secondkind ๐ ๐ฅ๐ฆ = ๐ { ๐ฅ,๐ฆ } = P โโ โ๏ธ ๐ง โ{ ๐ฅ,๐ฆ }โ X ๐ง โโ = P ( ๐ฅ โฉ ๐ฆ ) are probabilities of double intersections of eventsfrom { ๐ฅ, ๐ฆ } โ X , and probabilities of the secondkind ๐ ๐ ๐ = P โโ โ๏ธ ๐ฅ โ ๐ ๐ โ X ๐ฅ โโ are probabilities of ๐ -intersections of events from ๐ ๐ โ X , where | ๐ ๐ | = ๐ ; Definition 2 (set-phenomena of the set of probabil-ities of events from a s.e. and its phenomen-dom).
The ๐ -set of probabilities of events from X ห ๐ = { ๐ ๐ฅ : ๐ฅ โ X } also generates its X -phenomen-dom , the ๐ -totality ( ๐ | ห ๐ ) = {๏ธ ห ๐ ( ๐ | ๐// X ) , ๐ โ X }๏ธ , (2.2) OROBYEV composed of ๐ -sets in the form ห ๐ ( ๐ | ๐// X ) = {๏ธ ๐ ๐ง : ๐ง โ X ( ๐ | ๐ ) }๏ธ , and de๏ฌned for ๐ โ X as the ๐ -set of probabilitiesof events from ๐ -phenomenon X ( ๐ | ๐ ) of the s.e. X where for ๐ ๐ง โ ห ๐ ( ๐ | ๐// X ) ๐ ๐ง = {๏ธ ๐ ๐ฅ , ๐ง = ๐ฅ โ ๐, โ ๐ ๐ฅ ๐ง = ๐ฅ ๐ โ ๐ ( ๐ ) . In particular, for ๐ = X ห ๐ ( ๐ | X // X ) = { ๐ ๐ฅ : ๐ฅ โ X } = ห ๐. We denote by ๐ : โจ๏ธ ๐ฅ โ X [0 , ๐ฅ โ R +0 (2.3)a nonnegative bounded numerical function de-๏ฌned on the set-product [8], X -hypercube [0 , โ X = โจ๏ธ ๐ฅ โ X [0 , ๐ฅ . Arguments of ๐ form the ๐ -set ห ๐ค = { ๐ค ๐ฅ : ๐ฅ โ X } โ [0 , โ X which generates its own X -phenomenon-dom , the ๐ -totality ( ๐ | ห ๐ค ) = {๏ธ ห ๐ค ( ๐ | ๐// X ) , ๐ โ X }๏ธ (2.4)of ๐ -sets of arguments: ห ๐ค ( ๐ | ๐// X ) = {๏ธ ๐ค ๐ง : ๐ง โ X ( ๐ | ๐ ) }๏ธ (2.5)where for ๐ค ๐ง โ ห ๐ค ( ๐ | ๐// X ) ๐ค ๐ง = {๏ธ ๐ค ๐ฅ , ๐ง = ๐ฅ โ ๐, โ ๐ค ๐ฅ ๐ง = ๐ฅ ๐ โ ๐ ( ๐ ) . Let ฮจ X = {๏ธ ๐ โโ ๐ : [0 , โ X โ R +0 }๏ธ (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 ๐ โ ฮจ X is called normalized on the X -hypercube if for each ห ๐ค โ [0 , โ X โ๏ธ ๐ โ X ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = 1 , (2.7)i.e., the sum of its values on all the ๐ -sets of argu-ments from X -phenomenon-dom ( ๐ | ห ๐ค ) is one. Definition 4 ( a 1-function on the X -hypercube ). A function ๐ โ ฮจ X is called a on the X -hypercube if for all ห ๐ค โ [0 , โ X ๐ฅ - marginalequalities are satis๏ฌed for all ๐ฅ โ X : โ๏ธ ๐ฅ โ ๐ โ X ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = ๐ค ๐ฅ , (2.8)i.e., the sum of its values on ๐ฅ -halves of ๐ -sets ofarguments from the X -phenomenon-dom ( ๐ | ห ๐ค ) is ๐ค ๐ฅ .Denote by ฮจ X = โงโจโฉ ๐ โ ฮจ X : โ๏ธ ๐ โ X ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = 1; ห ๐ค โ [0 , โ X โซโฌโญ the family of functions, normalized on the X -hypercube; and by ฮจ X = โงโจโฉ ๐ โ ฮจ X : โ๏ธ ๐ฅ โ ๐ โ X ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = ๐ค ๐ฅ ; ห ๐ค โ [0 , โ X โซโฌโญ the family of 1-functions on the X -hypercube. properties of 1-functions on the { ๐ฅ, ๐ฆ } -square ). A strict inclusion is fair: ฮจ { ๐ฅ,๐ฆ } โ ฮจ { ๐ฅ,๐ฆ } . Proof . In other words, the lemma states: 1) if ๐ โ ฮจ { ๐ฅ,๐ฆ } is a 1-function on the { ๐ฅ, ๐ฆ } -square then ๐ โ ฮจ { ๐ฅ,๐ฆ } is a normalized function on the { ๐ฅ, ๐ฆ } -square; 2) among the normalized functions from ฮจ { ๐ฅ,๐ฆ } there is one which is not a 1-function. Butthis is obvious, as it is con๏ฌrmed by the followingsimple examples.First, indeed, for the doublet of events X = { ๐ฅ, ๐ฆ } bythe de๏ฌnition of a 1-function, we have ๐ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) + ๐ ( ๐ค ๐ฅ , โ ๐ค ๐ฆ ) = ๐ค ๐ฅ , (2.9) ๐ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) + ๐ (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) = ๐ค ๐ฆ , (2.10) ๐ (1 โ ๐ค ๐ฅ , โ ๐ค ๐ฆ ) + ๐ ( ๐ค ๐ฅ , โ ๐ค ๐ฆ ) = 1 โ ๐ค ๐ฆ , (2.11) ๐ (1 โ ๐ค ๐ฅ , โ ๐ค ๐ฆ ) + ๐ (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) = 1 โ ๐ค ๐ฅ . (2.12)The sums (2.9) and (2.12) as well as the sums (2.10)and (2.11) as a result give ๐ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) + ๐ ( ๐ค ๐ฅ , โ ๐ค ๐ฆ )++ ๐ (1 โ ๐ค ๐ฅ , โ ๐ค ๐ฆ ) + ๐ ( ๐ค ๐ฅ , โ ๐ค ๐ฆ ) = 1 , (2.13)i.e., ๐ โ ฮจ { ๐ฅ,๐ฆ } is a normalized function on the { ๐ฅ, ๐ฆ } -square.Second, the function (see its graph in Fig. 1 ) ๐ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) = ( ๐ค ๐ฅ + ๐ค ๐ฆ ) / In this ๏ฌgure 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 FAMEMSโ2015 C
ONFERENCE
Figure 1: The graph of the Cartesian representation of the normalizedfunction ๐ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) = ( ๐ค ๐ฅ + ๐ค ๐ฆ ) / on the { ๐ฅ, ๐ฆ } -square from ฮจ { ๐ฅ,๐ฆ } which is not a 1-function. is normalized on the { ๐ฅ, ๐ฆ } -square, since ๐ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) + ๐ ( ๐ค ๐ฅ , โ ๐ค ๐ฆ )+ ๐ (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) + ๐ (1 โ ๐ค ๐ฅ , โ ๐ค ๐ฆ ) == ( ๐ค ๐ฅ + ๐ค ๐ฆ ) / ๐ค ๐ฅ + 1 โ ๐ค ๐ฆ ) / โ ๐ค ๐ฅ + ๐ค ๐ฆ ) / โ ๐ค ๐ฅ + 1 โ ๐ค ๐ฆ ) / . However, it is not a 1-function, since ๐ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) + ๐ ( ๐ค ๐ฅ , โ ๐ค ๐ฆ ) = ( ๐ค ๐ฅ + ๐ค ๐ฆ ) / ๐ค ๐ฅ + 1 โ ๐ค ๐ฆ ) / ๐ค ๐ฅ / / ฬธ = ๐ค ๐ฅ ,๐ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) + ๐ (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) = ( ๐ค ๐ฅ + ๐ค ๐ฆ ) / โ ๐ค ๐ฅ + ๐ค ๐ฆ ) / ๐ค ๐ฆ / / ฬธ = ๐ค ๐ฆ . The lemma is proved.Of course, in the general case, for an arbitrary s.e. X the same lemma is ful๏ฌlled. Lemma 2 ( properties of 1-functions on the X -hypercube ). A strict inclusion is fair: ฮจ X โ ฮจ 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 ๐ โ ฮจ X on the X -hypercube [0 , โ X for each ห ๐ค โ [0 , โ X is represented in the form of | X | -set of the following functions: ๐ ( ห ๐ค ) = {๏ธ ๐ ๐ ( ห ๐ค ) , ๐ โ X }๏ธ == {๏ธ ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ , ๐ โ X }๏ธ . (2.14) Definition 5 (
Kopula ). The 1-functions K โ ฮจ X โ ฮจ 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 ห ๐ค โ [0 , โ X in theform of | X | -set of the following functions: K ( ห ๐ค ) = {๏ธ K ๐ ( ห ๐ค ) , ๐ โ X }๏ธ == {๏ธ K (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ , ๐ โ X }๏ธ . (2.15) Note 2 (characteristic properties of Kopula).
Each Kopula K has two characteristic properties1) Kopula is nonnegative : K (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ (cid:62) (2.16)for ๐ โ X , since by de๏ฌnition K โ ฮจ X ;2) Kopula ia satis๏ฌed ๐ฅ - marginal equalities : โ๏ธ ๐ฅ โ ๐ โ X K (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = ๐ค ๐ฅ (2.17)for ๐ฅ โ X , since by de๏ฌnition K โ ฮจ X ;From (2.17) by Lemma 2 a probabilistic normaliza-tion of the Kopula follows: โ๏ธ ๐ โ X K (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = 1 . (2.18)From (2.16) and (2.18) terrace-by-terrace probabilis-tic normalization of the Kopula follows: (cid:54) K (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ (cid:54) (2.19)for ๐ โ 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 ๐ = { ๐ ( ๐// X ) : ๐ โ X } be the e.p.d. ofthe 1st kind of the s.e. X with X -set of probabilitiesof marginal events ห ๐ = { ๐ ๐ฅ : ๐ฅ โ X } โ [0 , โ X . Thenthere is a | X | -Kopula K โ ฮจ X that de๏ฌnes a familyof e.p.d.โs of the 1st ๏ฌnd of the s.e. X . This familycontains the e.p.d. ๐ , when Kopulaโs arguments see the footnote 1 on page 78. OROBYEV coincide with ห ๐ . In other words, the such Kopulathat foe all ๐ โ X ๐ ( ๐// X ) = K ๐ (ห ๐ ) = K (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ . (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 ห ๐ โ [0 , โ X and any | X | -Kopula K โ ฮจ X , thefunction ๐ = { ๐ ( ๐// X ) : ๐ โ X } , de๏ฌned by formulas(2.20) for ๐ โ 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 ห ๐ . 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 ๐ = { ๐ ( ๐// X ) : ๐ โ X } of some s.e. X with the X -set ofmarginal probabilities ห ๐ = { ๐ ๐ฅ : ๐ฅ โ X } is de๏ฌned,then from properties of probabilities of the 1st kindit follows that for ๐ฅ โ X ๐ ๐ฅ = โ๏ธ ๐ฅ โ ๐ โ X ๐ ( ๐// X ) , (2.21)i.e., the function K , de๏ฌned by the e.p.d. of the1st kind ๐ and formulas (2.3), satis๏ฌes ๐ฅ -marginalequalities for ๐ฅ โ X : ๐ ๐ฅ = โ๏ธ ๐ฅ โ ๐ โ X K (ห ๐ ( ๐ | ๐// X ) ) (2.22)(required for being a 1-function:: K โ ฮจ X ) andserves as the | X | -Kopula.Second, if the function K is the | X | -Kopula, then byLemma 1: K โ ฮจ X โ ฮจ X , i.e., it is normalized and,consequently, by (2.20) the function ๐ is normalizedtoo: โ๏ธ ๐ โ X ๐ ( ๐// 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 ๐ฅ โ X . Therefore, the function ๐ is a e.p.d. of the 1stkind of the s.e. X with the X -set of marginal proba-bilities ห ๐ . 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 ๐ (2.24)be some set of its Kopulas. Let us prove that theirconvex combination K = ๐ โ๏ธ ๐ =1 ๐ผ ๐ K ๐ (2.25) (where, of course, ๐ผ + . . . + ๐ผ ๐ = 1 , ๐ผ ๐ (cid:62) , ๐ =1 , . . . , ๐ ) is also a Kopula. For this it su๏ฌces to provethat K is a 1-function. In other words, that for ๐ฅ โ X โ๏ธ ๐ฅ โ ๐ โ X K (ห ๐ ( ๐ | ๐// X ) ) = ๐ ๐ฅ . (2.26)Since each Kopula from the set (2.24) has a prop-erty of a 1-function, then for ๐ฅ โ X we get what isrequired: โ๏ธ ๐ฅ โ ๐ โ X K (ห ๐ ( ๐ | ๐// X ) ) == ๐ โ๏ธ ๐ =1 ๐ผ ๐ โ๏ธ ๐ฅ โ ๐ โ X K ๐ (ห ๐ ( ๐ | ๐// X ) ) == ๐ โ๏ธ ๐ =1 ๐ผ ๐ ๐ ๐ฅ = ๐ ๐ฅ . (2.27) Corollary 1 ( convex combination of Kopulas ). For every set of events X the space of 1-functions ฮจ 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.โs) (see [6]) and sets oftheir marginal probabilities, is unlikely to be ef-fective. However, in speci๏ฌc calculations at ๏ฌrst,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.eโs. 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 : ห ๐ = { ๐ ๐ฅ , ๐ฅ โ X } โ [0 , / โ X โ X -set of half-rare variables , ห ๐ ( ๐ | ๐// X ) โ [0 , / โ ๐ โ (1 / , โ X โ ๐ โ ๐ -renumbering ห ๐, ห ๐ค = { ๐ค ๐ฅ , ๐ฅ โ X } โ [0 , โ X โ X -set of free variables , ห ๐ค ( ๐ | ๐// X ) โ [0 , โ X โ ๐ -renumbering ห ๐ค, (2.28) Just remember [6], that the formula of ๐ -renumbering any X -set of probabilities of events has the form for ๐ โ X : ห ๐ ( ๐ | ๐// X ) = { ๐ ๐ฅ , ๐ฅ โ ๐ } + { โ ๐ ๐ฅ , ๐ฅ โ X โ ๐ } . HE XIV FAMEMSโ2015 C
ONFERENCE and always interpreting them as probabilities ofevents. In particular, for the half-rare doublet X = { ๐ฅ, ๐ฆ } we have: ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ } โ [0 , / ๐ฅ โ [0 , / ๐ฆ โ X -set of half-rare variables ,๐ ๐ฅ๐ฆ ( ๐ ๐ฅ , ๐ ๐ฆ ) โ [0 , min { ๐ ๐ฅ , ๐ ๐ฆ } ] โ half-rare function of half-rare variables , ห ๐ค = { ๐ค ๐ฅ , ๐ค ๐ฆ } โ [0 , ๐ฅ โ [0 , ๐ฆ โ X -set of free variables ,๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) โ [0 , min { ๐ค ๐ฅ , ๐ค ๐ฆ } ] โ free function of free variables . (2.29) Note 3 (phenomenon replacement between half-rare andfree variables).
For every ๐ โ X phenomenonreplacement of half-rare variables ห ๐ โ [0 , / โ X byfree variables ห ๐ค โ [0 , โ X and vise-versa is de๏ฌnedfor ๐ โ X by mutually inverse formulas of theset-phenomenon transformation of the form: ห ๐ = ห ๐ ( ๐ | X // X ) == โงโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโฉ ห ๐ค ( ๐ |โ // X ) , ๐ค ๐ฅ > / , ๐ฅ โ X ,. . . , . . . ห ๐ค ( ๐ | ๐// X ) , ๐ค ๐ฅ (cid:54) / , ๐ฅ โ ๐,๐ค ๐ฅ > / , ๐ฅ โ X โ ๐,. . . , . . . ห ๐ค ( ๐ | X // X ) , ๐ค ๐ฅ (cid:54) / , ๐ฅ โ X (2.30)where for ๐ฅ โ X the following agreement is alwaysaccepted (see, for example, paragraph 11.7): ๐ ๐ฅ = {๏ธ ๐ค ๐ฅ , ๐ค ๐ฅ (cid:54) / , โ ๐ค ๐ฅ , ๐ค ๐ฅ > / . (2.31) Letโs construct the -Kopula K โ ฮจ X of a familyof e.p.d.โs of monoplet of events X = { ๐ฅ } with thee.p.d. of the 1st kind {๏ธ ๐ ( ๐// { ๐ฅ } ) , ๐ โ { ๐ฅ } }๏ธ = { ๐ ( โ // { ๐ฅ } ) , ๐ ( ๐ฅ// { ๐ฅ } ) } and { ๐ฅ } -monoplet of marginal probabilities { ๐ ๐ฅ } ,where ๐ ๐ฅ = P ( ๐ฅ ) = ๐ ( ๐ฅ// { ๐ฅ } ) . In other words, letโs construct a 1-function on theunit X -segment, i.e., a such nonnegative boundednumerical function K : [0 , โ [0 , , that for ๐ฅ โ X โ๏ธ ๐ฅ โ ๐ โ X K (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = ๐ค ๐ฅ . Since for ๐ โ X = { ๐ฅ } K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ } ) )๏ธ = {๏ธ K (1 โ ๐ค ๐ฅ ) , ๐ = โ , K ( ๐ค ๐ฅ ) , ๐ = { ๐ฅ } , then a marginal and global normalization of thefunction K are written as: K ( ๐ค ๐ฅ ) = ๐ค ๐ฅ , K ( ๐ค ๐ฅ ) + K (1 โ ๐ค ๐ฅ ) = 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 = { ๐ฅ } is de๏ฌned for free variables ห ๐ค = { ๐ค ๐ฅ } โ [0 , ๐ฅ by a one formula: K ( ห ๐ค ) = K ( ๐ค ๐ฅ ) = ๐ค ๐ฅ , (2.33)which provides two values on each ( ๐ | ห ๐ค ) -penomenon-dom by โfreeโ formulas: K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ } ) )๏ธ == {๏ธ K (1 โ ๐ค ๐ฅ ) = 1 โ ๐ค ๐ฅ , ๐ = โ , K ( ๐ค ๐ฅ ) = ๐ค ๐ฅ , ๐ = { ๐ฅ } . (2.34)and the e.p.d. of the 1st kind of this monoplet with { ๐ฅ } -monoiplet of probabilities of events ห ๐ = { ๐ ๐ฅ } โ [0 , ๐ฅ are de๏ฌned for half-rare variables by the 1-Kopula (2.33) for ๐ โ { ๐ฅ } by exactly the same โhalf-rareโ formulas: ๐ ( ๐// { ๐ฅ } ) = K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ } ) )๏ธ == {๏ธ K (1 โ ๐ ๐ฅ ) = 1 โ ๐ ๐ฅ , ๐ = โ , K ( ๐ ๐ฅ ) = ๐ ๐ฅ , ๐ = { ๐ฅ } . (2.35) Letโs construct an example of K โ ฮจ X offamilies of a doublet of events X = { ๐ฅ, ๐ฆ } , in otherwords, letโs construct on the unit { ๐ฅ, ๐ฆ } -square thesuch nonnegative bounded numerical functions K : [0 , โ{ ๐ฅ,๐ฆ } โ [0 , , that for all ๐ง โ { ๐ฅ, ๐ฆ } โ๏ธ ๐ง โ ๐ โ{ ๐ฅ,๐ฆ } K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ = ๐ค ๐ง . Since each 2-set of arguments ห ๐ค โ [0 , ๐ฅ โ [0 , ๐ฆ gen-erates { ๐ฅ,๐ฆ } -phenomenon-dom ( ๐ | ห ๐ค ) = { ห ๐ค, ห ๐ค ( ๐ |{ ๐ฅ } ) , ห ๐ค ( ๐ |{ ๐ฆ } ) , ห ๐ค ( ๐ |โ ) } , (2.36)composed from forth its set-phenomena ห ๐ค = ห ๐ค ( ๐ |{ ๐ฅ,๐ฆ } // { ๐ฅ,๐ฆ } ) = { ๐ค ๐ฅ , ๐ค ๐ฆ } , ห ๐ค ( ๐ |{ ๐ฅ } // { ๐ฅ,๐ฆ } ) = { ๐ค ๐ฅ , โ ๐ค ๐ฆ } , ห ๐ค ( ๐ |{ ๐ฅ } // { ๐ฅ,๐ฆ } ) = { โ ๐ค ๐ฅ , ๐ค ๐ฆ } , ห ๐ค ( ๐ |โ // { ๐ฅ,๐ฆ } ) = { โ ๐ค ๐ฅ , โ ๐ค ๐ฆ } , (2.37) OROBYEV then K ( ห ๐ค ) = K ( ๐ค ๐ฅ , ๐ค ๐ฆ ) , K (๏ธ ห ๐ค ( ๐ |{ ๐ฅ } // { ๐ฅ,๐ฆ } ) )๏ธ = K ( ๐ค ๐ฅ , โ ๐ค ๐ฆ ) , K (๏ธ ห ๐ค ( ๐ |{ ๐ฆ } // { ๐ฅ,๐ฆ } ) )๏ธ = K (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) K (๏ธ ห ๐ค ( ๐ |โ // { ๐ฅ,๐ฆ } ) )๏ธ = K (1 โ ๐ค ๐ฅ , โ ๐ค ๐ฆ ) , and normalizations for every ห ๐ค โ [0 , โ{ ๐ฅ,๐ฆ } arewritten as: K ( ๐ค ๐ฅ , ๐ค ๐ฆ ) + K ( ๐ค ๐ฅ , โ ๐ค ๐ฆ ) = ๐ค ๐ฅ , K ( ๐ค ๐ฅ , ๐ค ๐ฆ ) + K (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) = ๐ค ๐ฆ , K ( ๐ค ๐ฅ , ๐ค ๐ฆ ) + K ( ๐ค ๐ฅ , โ ๐ค ๐ฆ )++ K (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) + K (1 โ ๐ค ๐ฅ , โ ๐ค ๐ฆ ) = 1 . The e.p.d. of the 1st kind of doublet of events isde๏ฌned by the 2-Kopula for ๐ โ { ๐ฅ, ๐ฆ } in half-rarevariables by general formulas: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโจโชโชโชโฉ K (1 โ ๐ ๐ฅ , โ ๐ ๐ฆ ) , ๐ = โ , K ( ๐ ๐ฅ , โ ๐ ๐ฆ ) , ๐ = { ๐ฅ } . K (1 โ ๐ ๐ฅ , ๐ ๐ฆ ) , ๐ = { ๐ฆ } , K ( ๐ ๐ฅ , ๐ ๐ฆ ) , ๐ = { ๐ฅ, ๐ฆ } , = โงโชโชโชโจโชโชโชโฉ โ ๐ ๐ฅ โ ๐ ๐ฆ + ๐ ๐ฅ๐ฆ (ห ๐ ) , ๐ = โ ,๐ ๐ฅ โ ๐ ๐ฅ๐ฆ (ห ๐ ) , ๐ = { ๐ฅ } .๐ ๐ฆ โ ๐ ๐ฅ๐ฆ (ห ๐ ) , ๐ = { ๐ฆ } ,๐ ๐ฅ๐ฆ (ห ๐ ) , ๐ = { ๐ฅ, ๐ฆ } , (2.38)where ๐ ๐ฅ๐ฆ (ห ๐ ) 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: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ ๐ค ๐ฅ + ๐ค ๐ฆ โ ๐ค ๐ฅ๐ฆ (1 โ ๐ค ๐ฅ , โ ๐ค ๐ฆ ) ,๐ค ๐ฅ > / , ๐ค ๐ฆ > / โ ๐ = โ ,๐ค ๐ฅ โ ๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , โ ๐ค ๐ฆ ) ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ > / โ ๐ = { ๐ฅ } ,๐ค ๐ฆ โ ๐ค ๐ฅ๐ฆ (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) ,๐ค ๐ฅ > / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฆ } ,๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฅ, ๐ฆ } . (2.39) The simplest example of a 1-function on a { ๐ฅ, ๐ฆ } -square is the so-called independent 2-Kopula , which for free variables ห ๐ค โ [0 , โ{ ๐ฅ,๐ฆ } is de๏ฌned by theformula: K ( ห ๐ค ) = ๐ค ๐ฅ ๐ค ๐ฆ . (2.40)This provides it on each ( ๐ | ห ๐ค ) -phenomenon the fol-lowing four values: K (๏ธ ห ๐ค ( ๐ |{ ๐ฅ,๐ฆ } // { ๐ฅ,๐ฆ } ) )๏ธ = ๐ค ๐ฅ ๐ค ๐ฆ , K (๏ธ ห ๐ค ( ๐ |{ ๐ฅ } // { ๐ฅ,๐ฆ } ) )๏ธ = ๐ค ๐ฅ (1 โ ๐ค ๐ฆ ) , K (๏ธ ห ๐ค ( ๐ |{ ๐ฆ } // { ๐ฅ,๐ฆ } ) )๏ธ = (1 โ ๐ค ๐ฅ ) ๐ค ๐ฆ , K (๏ธ ห ๐ค ( ๐ |โ // { ๐ฅ,๐ฆ } ) )๏ธ = (1 โ ๐ค ๐ฅ )(1 โ ๐ค ๐ฆ ) . (2.41)Indeed, the so-de๏ฌned independent 2-Kopula is a 1-function because โ๏ธ ๐ฅ โ ๐ โ{ ๐ฅ,๐ฆ } K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ = ๐ค ๐ฅ ๐ค ๐ฆ + ๐ค ๐ฅ (1 โ ๐ค ๐ฆ ) = ๐ค ๐ฅ , โ๏ธ ๐ฆ โ ๐ โ{ ๐ฅ,๐ฆ } K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ = ๐ค ๐ฅ ๐ค ๐ฆ + (1 โ ๐ค ๐ฅ ) ๐ค ๐ฆ = ๐ค ๐ฆ . The e.p.d. of the 1st kind of doublet of independentevents with the { ๐ฅ, ๐ฆ } -set of probabilities of events ห ๐ is de๏ฌned by four values of the independent 2-Kopula (2.40) on its ( ๐ | ห ๐ ) -penomenon-dom by gen-eral formulas in half-rare variables (see Fig. 2), i.e.,for ๐ โ { ๐ฅ, ๐ฆ } : ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโจโชโชโชโฉ (1 โ ๐ ๐ฅ )(1 โ ๐ ๐ฆ ) , ๐ = โ ,๐ ๐ฅ (1 โ ๐ ๐ฆ ) , ๐ = { ๐ฅ } . (1 โ ๐ ๐ฅ ) ๐ ๐ฆ , ๐ = { ๐ฆ } ,๐ ๐ฅ ๐ ๐ฆ , ๐ = { ๐ฅ, ๐ฆ } . (2.42) With the phenomenal substitution (2.30) half-rare2-Kopula as a function of the free variables takesthe equivalent form: K ( ห ๐ค ) = โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ ๐ค ๐ฅ๐ฆ (๏ธ ห ๐ค ( ๐ | X // X ) )๏ธ , ห ๐ค โ [0 , / ๐ฅ โ [0 , / ๐ฆ ,๐ค ๐ฅ โ ๐ค ๐ฅ๐ฆ (๏ธ ห ๐ค ( ๐ |{ ๐ฅ } // X ) )๏ธ , ห ๐ค โ [0 , / ๐ฅ โ (1 / , ๐ฆ ,๐ค ๐ฆ โ ๐ค ๐ฅ๐ฆ (๏ธ ห ๐ค ( ๐ |{ ๐ฆ } // X ) )๏ธ , ห ๐ค โ (1 / , ๐ฅ โ [0 , / ๐ฆ ,๐ค ๐ฅ + ๐ค ๐ฆ โ ๐ค ๐ฅ๐ฆ (๏ธ ห ๐ค ( ๐ |โ // X ) )๏ธ , ห ๐ค โ (1 / , ๐ฅ โ (1 / , ๐ฆ . (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 FAMEMSโ2015 C
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Figure 2: Graphs of Cartesian representation of the 2-Kopula of a familyof e.p.d.โs of an independent half-rare doublet of events { ๐ฅ, ๐ฆ } ; probabil-ities of the 1st kind are marked by different colors : ๐ ( ๐ฅ๐ฆ ) (aqua), ๐ ( ๐ฅ ) (lime), ๐ ( ๐ฆ ) (yellow) ๐ ( โ ) (fuchsia). calculations: K ( ห ๐ค ) = โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ ๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) , ห ๐ค โ [0 , / ๐ฅ โ [0 , / ๐ฆ ,๐ค ๐ฅ โ ๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , โ ๐ค ๐ฆ ) , ห ๐ค โ [0 , / ๐ฅ โ (1 / , ๐ฆ ,๐ค ๐ฆ โ ๐ค ๐ฅ๐ฆ (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) , ห ๐ค โ (1 / , ๐ฅ โ [0 , / ๐ฆ ,๐ค ๐ฅ + ๐ค ๐ฆ โ ๐ค ๐ฅ๐ฆ (1 โ ๐ค ๐ฅ , โ ๐ค ๐ฆ ) , ห ๐ค โ (1 / , ๐ฅ โ (1 / , ๐ฆ ; (2.44)and again with restrictions in the form of the famil-iar โhumanโ inequalities : K ( ห ๐ค ) = โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ ๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ (cid:54) / ,๐ค ๐ฅ โ ๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , โ ๐ค ๐ฆ ) ,๐ค ๐ฅ (cid:54) / , / < ๐ค ๐ฆ ,๐ค ๐ฆ โ ๐ค ๐ฅ๐ฆ (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) , / < ๐ค ๐ฅ , ๐ค ๐ฆ (cid:54) / ,๐ค ๐ฅ + ๐ค ๐ฆ โ ๐ค ๐ฅ๐ฆ (1 โ ๐ค ๐ฅ , โ ๐ค ๐ฆ ) , / < ๐ค ๐ฅ , / < ๐ค ๐ฆ (2.45) probabilistic normalization: โ (cid:54) ... (cid:54) โ is assumed by de-fault. where (cid:54) ๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) (cid:54) min { ๐ค ๐ฅ , ๐ค ๐ฆ } , (cid:54) ๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , โ ๐ค ๐ฆ ) (cid:54) min { ๐ค ๐ฅ , โ ๐ค ๐ฆ } , (cid:54) ๐ค ๐ฅ๐ฆ (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) (cid:54) min { โ ๐ค ๐ฅ , ๐ค ๐ฆ } , (cid:54) ๐ค ๐ฅ๐ฆ (1 โ ๐ค ๐ฅ , โ ๐ค ๐ฆ ) (cid:54) min { โ ๐ค ๐ฅ , โ ๐ค ๐ฆ } (2.46)are the Frยดechet inequalities for ๐ค ๐ฅ๐ฆ as the half-rare probability of double intersection of half-rareevents: either half-rare events ๐ฅ or ๐ฆ , or their half-rare complements, when events ๐ฅ or ๐ฆ 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 ๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) = ๐ค ๐ฅ ๐ค ๐ฆ , which corre-sponds to the probability of double intersectionof independent events ๐ฅ and ๐ฆ happening withprobabilities ๐ค ๐ฅ and ๐ค ๐ฆ , and means, of course, thatthe all the following four equations are satis๏ฌed: ๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) = ๐ค ๐ฅ ๐ค ๐ฆ ,๐ค ๐ฅ๐ฆ ( ๐ค ๐ฅ , โ ๐ค ๐ฆ ) = ๐ค ๐ฅ (1 โ ๐ค ๐ฆ ) ,๐ค ๐ฅ๐ฆ (1 โ ๐ค ๐ฅ , ๐ค ๐ฆ ) = (1 โ ๐ค ๐ฅ ) ๐ค ๐ฆ ,๐ค ๐ฅ๐ฆ (1 โ ๐ค ๐ฅ , โ ๐ค ๐ฆ ) = (1 โ ๐ค ๐ฅ )(1 โ ๐ค ๐ฆ ); from (2.45) it follows that a half-rare 2-Kopula fromfree variables of the family of e.p.d.โs of the inde-pendent doublet of events X = { ๐ฅ, ๐ฆ } with X -sets offree marginal probabilities ห ๐ค = { ๐ค ๐ฅ , ๐ค ๐ฆ } โ [0 , โ X has the same view on all ( ๐ | ห ๐ค ) -phenomenon-doms: K ( ห ๐ค ) = ๐ค ๐ฅ ๐ค ๐ฆ . (2.47) An example of a 1-function on a { ๐ฅ, ๐ฆ } -square is theso-called upper 2-Kopula of Frยดechet , which suggeststhe probabilities of a double intersection to be itsupper Frยดechet boundary. In other words, the onlyfunctional free parameter in (2.39) is: ๐ค ๐ฅ๐ฆ ( ห ๐ค ) = ๐ค + ๐ฅ๐ฆ ( ห ๐ค ) = min { ๐ค ๐ฅ , ๐ค ๐ฆ } . (2.48) OROBYEV The upper 2-Kopula of Frยดechet from free variables ห ๐ค โ [0 , โ{ ๐ฅ,๐ฆ } is de๏ฌned by the formulas: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ ๐ค ๐ฅ + ๐ค ๐ฆ โ { โ ๐ค ๐ฅ , โ ๐ค ๐ฆ } ,๐ค ๐ฅ > / , ๐ค ๐ฆ > / โ ๐ = โ ,๐ค ๐ฅ โ min { ๐ค ๐ฅ , โ ๐ค ๐ฆ } ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ > / โ ๐ = { ๐ฅ } ,๐ค ๐ฆ โ min { โ ๐ค ๐ฅ , ๐ค ๐ฆ } ,๐ค ๐ฅ > / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฆ } , min { ๐ค ๐ฅ , ๐ค ๐ฆ } ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฅ, ๐ฆ } . (2.49)After simple transformations, these formulas pro-vide the upper 2-Kopula of Frยดechet on each ( ๐ | ห ๐ค ) -phenomenon-dom the following four values of freevariables: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ min { ๐ค ๐ฅ , ๐ค ๐ฆ } ,๐ค ๐ฅ > / , ๐ค ๐ฆ > / โ ๐ = โ , max { , ๐ค ๐ฅ + ๐ค ๐ฆ โ } ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ > / โ ๐ = { ๐ฅ } , max { , ๐ค ๐ฅ + ๐ค ๐ฆ โ } ,๐ค ๐ฅ > / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฆ } , min { ๐ค ๐ฅ , ๐ค ๐ฆ } ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฅ, ๐ฆ } , = โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ ๐ค + ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) ,๐ค ๐ฅ > / , ๐ค ๐ฆ > / โ ๐ = โ ,๐ค โ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ > / โ ๐ = { ๐ฅ } ,๐ค โ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) ,๐ค ๐ฅ > / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฆ } ,๐ค + ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฅ, ๐ฆ } , (2.50)which, as can be seen, are de๏ฌned by the upper andlower Frยดechet 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: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโชโชโจโชโชโชโชโชโฉ min { โ ๐ ๐ฅ , โ ๐ ๐ฆ } , ๐ = โ , max { , ๐ ๐ฅ โ ๐ ๐ฆ } , ๐ = { ๐ฅ } , max { , ๐ ๐ฆ โ ๐ ๐ฅ } , ๐ = { ๐ฆ } , min { ๐ ๐ฅ , ๐ ๐ฆ } , ๐ = { ๐ฅ, ๐ฆ } . (2.51) If ๐ ๐ฅ (cid:62) ๐ ๐ฆ , this formula takes the form: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโชโชโจโชโชโชโชโชโฉ โ ๐ ๐ฅ , ๐ = โ ,๐ ๐ฅ โ ๐ ๐ฆ , ๐ = { ๐ฅ } , , ๐ = { ๐ฆ } ,๐ ๐ฆ , ๐ = { ๐ฅ, ๐ฆ } . (2.52)And if ๐ ๐ฅ < ๐ ๐ฆ , then this formula takes the form: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโชโชโจโชโชโชโชโชโฉ โ ๐ ๐ฆ , ๐ = โ , , ๐ = { ๐ฅ } ,๐ ๐ฆ โ ๐ ๐ฅ , ๐ = { ๐ฆ } ,๐ ๐ฆ , ๐ = { ๐ฅ, ๐ฆ } . (2.53)So the de๏ฌnite upper 2-Kopula of Frยดechet is indeeda 1-function, due to the fact that when ๐ค ๐ฅ (cid:62) ๐ค ๐ฆ โ๏ธ ๐ฅ โ ๐ โ{ ๐ฅ,๐ฆ } K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ = ๐ค ๐ฆ + ( ๐ค ๐ฅ โ ๐ค ๐ฆ ) = ๐ค ๐ฅ , โ๏ธ ๐ฆ โ ๐ โ{ ๐ฅ,๐ฆ } K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ = ๐ค ๐ฆ + 0 = ๐ค ๐ฆ , (2.54)and when ๐ค ๐ฅ < ๐ค ๐ฆ โ๏ธ ๐ฅ โ ๐ โ{ ๐ฅ,๐ฆ } K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ = ๐ค ๐ฅ + 0 = ๐ค ๐ฅ , โ๏ธ ๐ฆ โ ๐ โ{ ๐ฅ,๐ฆ } K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ = ๐ค ๐ฅ + ( ๐ค ๐ฆ โ ๐ค ๐ฅ ) = ๐ค ๐ฆ . (2.55) A once more example of a 1-function on a { ๐ฅ, ๐ฆ } -square is the so-called lower 2-Kopula of Frยดechet ,which suggests the probabilities of a double inter-section to be its upper Frยดechet boundary. In otherwords, the only functional free parameter in (2.39)is: ๐ค ๐ฅ๐ฆ ( ห ๐ค ) = ๐ค โ ๐ฅ๐ฆ ( ห ๐ค ) = max { , ๐ค ๐ฅ + ๐ค ๐ฆ โ } . (2.56) HE XIV FAMEMSโ2015 C
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Figure 3: Graphs of the Cartesian representation of the upper 2-Kopula ofFrยดechet of a family of e.p.d.โs of half-rare doublet of events { ๐ฅ, ๐ฆ } ; proba-bilities of the 1st kind are marked by different colors: ๐ ( ๐ฅ๐ฆ ) (aqua), ๐ ( ๐ฅ ) (lime), ๐ ( ๐ฆ ) (yellow) ๐ ( โ ) (fuchsia).Figure 4: Graphs of the Cartesian representation of the lower 2-Kopula ofFrยดechet of a family of e.p.d.โs of half-rare doublet of events { ๐ฅ, ๐ฆ } ; proba-bilities of the 1st kind are marked by different colors: ๐ ( ๐ฅ๐ฆ ) (aqua), ๐ ( ๐ฅ ) (lime), ๐ ( ๐ฆ ) (yellow) ๐ ( โ ) (fuchsia). The lower 2-Kopula of Frยดechet from free variables ห ๐ค โ [0 , โ{ ๐ฅ,๐ฆ } is de๏ฌned by the formulas: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ ๐ค ๐ฅ + ๐ค ๐ฆ โ { , โ ๐ค ๐ฅ โ ๐ค ๐ฆ } ,๐ค ๐ฅ > / , ๐ค ๐ฆ > / โ ๐ = โ ,๐ค ๐ฅ โ max { , ๐ค ๐ฅ โ ๐ค ๐ฆ } ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ > / โ ๐ = { ๐ฅ } ,๐ค ๐ฆ โ max { , ๐ค ๐ฆ โ ๐ค ๐ฅ } ,๐ค ๐ฅ > / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฆ } , max { , ๐ค ๐ฅ + ๐ค ๐ฆ โ } ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฅ, ๐ฆ } . (2.57) After simple transformations, these formulas pro-vide the lower 2-Kopula of Frยดechet on each ( ๐ | ห ๐ค ) -phenomenon-dom the following four values of freevariables: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ max { , ๐ค ๐ฅ + ๐ค ๐ฆ โ } ,๐ค ๐ฅ > / , ๐ค ๐ฆ > / โ ๐ = โ , min { ๐ค ๐ฅ , ๐ค ๐ฆ } ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ > / โ ๐ = { ๐ฅ } , min { ๐ค ๐ฅ , ๐ค ๐ฆ } ,๐ค ๐ฅ > / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฆ } , max { , ๐ค ๐ฅ + ๐ค ๐ฆ โ } ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฅ, ๐ฆ } , = โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ ๐ค โ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) ,๐ค ๐ฅ > / , ๐ค ๐ฆ > / โ ๐ = โ ,๐ค + ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ > / โ ๐ = { ๐ฅ } ,๐ค + ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) ,๐ค ๐ฅ > / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฆ } ,๐ค โ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) ,๐ค ๐ฅ (cid:54) / , ๐ค ๐ฆ (cid:54) / โ ๐ = { ๐ฅ, ๐ฆ } , (2.58)which, as can be seen, are also de๏ฌned by the upperand lower Frยดechet 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: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโชโชโจโชโชโชโชโชโฉ max { , โ ๐ ๐ฅ โ ๐ ๐ฆ } , ๐ = โ , min { ๐ ๐ฅ , โ ๐ ๐ฆ } , ๐ = { ๐ฅ } , min { โ ๐ ๐ฅ , ๐ ๐ฆ } , ๐ = { ๐ฆ } , max { , ๐ ๐ฅ + ๐ ๐ฆ โ } , ๐ = { ๐ฅ, ๐ฆ } , = โงโชโชโชโชโชโจโชโชโชโชโชโฉ โ ๐ ๐ฅ โ ๐ ๐ฆ , ๐ = โ ,๐ ๐ฅ , ๐ = { ๐ฅ } ,๐ ๐ฆ , ๐ = { ๐ฆ } , , ๐ = { ๐ฅ, ๐ฆ } , (2.59)So the de๏ฌnite lower 2-Kopula of Frยดechet is indeeda 1-function, due to the fact that for all half-rarevariables โ๏ธ ๐ฅ โ ๐ โ{ ๐ฅ,๐ฆ } K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ = ๐ ๐ฅ + 0 = ๐ ๐ฅ , โ๏ธ ๐ฆ โ ๐ โ{ ๐ฅ,๐ฆ } K (๏ธ ห ๐ค ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ = ๐ ๐ฆ + 0 = ๐ ๐ฆ . (2.60) OROBYEV { ๐ฅ, ๐ฆ } with functional weight parameter (2.63) inthe formula (2.61); probabilities of the 1st kind are marked by differentcolors: ๐ ( ๐ฅ๐ฆ ) (aqua), ๐ ( ๐ฅ ) (lime), ๐ ( ๐ฆ ) (yellow) ๐ ( โ ) (fuchsia). A rather general example of a 1-function on a { ๐ฅ, ๐ฆ } -square is the convex combinations of the upper,lower, and independent 2-Kopulas ofwise Frยดechet ,which propose the probabilities of a pair inter-section to become a convex combination (this isallowed by the lemma 3) of its upper and lowerFrยดechet 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 Frยดechet can be ensured by the uniquefunctional free parameter ๐ค ๐ฅ๐ฆ ( ห ๐ค ) in (2.39), in whichthe probability of double intersection is computedby the following formula: ๐ค ๐ฅ๐ฆ ( ห ๐ค ) == (1 โ ๐ผ ) / ๐ค โ ๐ฅ๐ฆ ( ห ๐ค ) + (1+ ๐ผ ) / ๐ค + ๐ฅ๐ฆ ( ห ๐ค ) (2.61)where ๐ผ = ๐ผ ( ห ๐ค ) โ [ โ , is an arbitrary function on [0 , โ{ ๐ฅ,๐ฆ } with values from [ โ , , and ๐ค โ ๐ฅ๐ฆ ( ห ๐ค ) = max { , ๐ค ๐ฅ + ๐ค ๐ฆ โ } ,๐ค + ๐ฅ๐ฆ ( ห ๐ค ) = min { ห ๐ค } (2.62) are the lower and upper Frยดechet-boundaries ofprobability of double intersection.For ๐ผ = โ , the probability ๐ค ๐ฅ๐ฆ ( ห ๐ค ) = ๐ค โ ๐ฅ๐ฆ ( ห ๐ค ) co-incides with the lower Frยดechet boundary of half-rare marginal probabilities; for ๐ผ = 1 , the prob-ability ๐ค ๐ฅ๐ฆ ( ห ๐ค ) = ๐ค + ๐ฅ๐ฆ ( ห ๐ค ) coincides with the upperFrยดechet boundary of marginal probabilities. Un-fortunately, these are the properties of a convexcombination such that for ๐ผ = 0 the probability ofdouble intersection is equal to half of the sum ofits lower and upper Frยดechet boundaries: ๐ค ๐ฅ๐ฆ ( ห ๐ค ) =( ๐ค โ ๐ฅ๐ฆ ( ห ๐ค ) + ๐ค + ๐ฅ๐ฆ ( ห ๐ค )) / (see Figure6), and not an inde-pendent 2-Kopula, no matter how much we wantit. This โblunderโ 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 [ โ , : ๐ผ = ๐ผ ( ห ๐ค ) = sin(15( ๐ค ๐ฅ โ ๐ค ๐ฆ )) . (2.63) Figure 6: Graphs of the Cartesian representation of the convex combi-nation of upper and lower 2-Kopulas of Frยดechet of a family of e.p.d.โs ofhalf-rare doublet of events { ๐ฅ, ๐ฆ } with the constant functional weight pa-rameter ๐ผ ( ห ๐ค ) = 0 in the formula (2.61); probabilities of the 1st kind aremarked by different colors: ๐ ( ๐ฅ๐ฆ ) (aqua), ๐ ( ๐ฅ ) (lime), ๐ ( ๐ฆ ) (yellow) ๐ ( โ ) (fuchsia). We construct two convex combinations of the theindependent 2-copula and the lower and upper 2-Kopulas of Frยดechet. The conjugation of these two HE XIV FAMEMSโ2015 C
ONFERENCE convex combinations can be ensured by the uniquefunctional free parameter ๐ค ๐ฅ๐ฆ ( ห ๐ค ) in (2.39) by thefollowing conjugation formula for two convex com-binations: ๐ค ๐ฅ๐ฆ ( ห ๐ค ) == โงโชโชโชโชโชโจโชโชโชโชโชโฉ ๐ค + ๐ฅ๐ฆ ( ห ๐ค ) max { ห ๐ค } (1+ ๐ผ ) ,๐ผ (cid:54) ๐ค + ๐ฅ๐ฆ ( ห ๐ค ) (๏ธ max { ห ๐ค } (1 โ ๐ผ )+ ๐ผ )๏ธ ,๐ผ > , (2.64)where ๐ผ = ๐ผ ( ห ๐ค ) โ [ โ , is an arbitrary function on [0 , โ{ ๐ฅ,๐ฆ } with values from [ โ , , and ๐ค + ๐ฅ๐ฆ ( ห ๐ค ) = min { ห ๐ค } (2.65)is the upper Frยดechet-boundary of probability ofdouble intersection. For ๐ผ = 0 , the probability ๐ค ๐ฅ๐ฆ ( ห ๐ค ) = min { ห ๐ค } max { ห ๐ค } = ๐ค ๐ฅ ๐ค ๐ฆ coincides with theprobability of double intersection of independentevents; for ๐ผ = โ , the probability ๐ค ๐ฅ๐ฆ ( ห ๐ค ) = 0 coin-cides with the lower Frยดechet-boundary of half-raremarginal probabilities; for ๐ผ = 1 , the probability ๐ค ๐ฅ๐ฆ ( ห ๐ค ) = ๐ค + ๐ฅ๐ฆ ( ห ๐ค ) coincides with the upper Frยดechet-boundary of marginal probabilities.In Fig. 5 it is a graph of this 2-Kopula for the sameweight function with values from [ โ , as in theprevious example. ๐ผ = ๐ผ ( ห ๐ค ) = sin(15( ๐ค ๐ฅ โ ๐ค ๐ฆ )) . (2.66) Definition 6 (inserted s.e.โs).
For each pais ofs.e.โs ๐ณ and ๐ด with the joint e.p.d. { ๐ ( ๐ + ๐ // ๐ณ + ๐ด ) } , ๐ โ ๐ณ , ๐ โ ๐ด} (3.1)for every ๐ โ ๐ด the ๐ -inserted s.e.โs, generatedby ๐ณ , in the frame s.e. ๐ด are s.e.โs, which are de-noted by ๐ณ ( โฉ ๐// ๐ด ) , and de๏ฌned as the following M-intersection : ๐ณ ( โฉ ๐// ๐ด ) = ๐ณ ( โฉ ) { ter ( ๐ // ๐ด ) } == { ๐ฅ โฉ ter ( ๐ // ๐ด ) , ๐ฅ โ ๐ณ } (3.2)and have the e.p.d., which coincides with the pro-jection of the joint e.p.d. (3.1) for ๏ฌxed ๐ โ ๐ด andevery ๐ โ ๐ณ : ๐ ( ๐ ( โฉ ) { ter ( ๐ // ๐ด ) } ) = ๐ ( ๐ + ๐ // ๐ณ + ๐ด ) } . (3.3) M-intersection is an intersection by Minkowski.
Definition 7 (event-probabilistic pseudo-distribution of an inserted s.e.).
For each ๐ โ ๐ด the ๐ -inserted s.e. ๐ณ ( โฉ ๐// ๐ด ) = ๐ณ ( โฉ ) { ter ( ๐ // ๐ด ) } (3.4)with the e.p.d. (3.3) has the event-probabilistic ๐ -pseudo-distribution , which is de๏ฌned as a set ofprobabilitieso of terraced events that coincide withprobabilities from the e.p.d. (3.3) for all ๐ โ ๐ณ ex-cepting ๐ = โ : ๐ ( ๐ ) ( ๐ + ๐ // ๐ณ + ๐ด ) == {๏ธ ๐ ( ๐ + ๐ // ๐ณ + ๐ด ) , ๐ ฬธ = โ ,๐ ( ๐ // ๐ณ + ๐ด ) โ ๐ ( ๐ // ๐ด ) , ๐ = โ , = โงโชโจโชโฉ ๐ ( ๐ + ๐ // ๐ณ + ๐ด ) , ๐ ฬธ = โ ,๐ ( ๐ // ๐ด ) โ โ๏ธ ๐ ฬธ = โ ๐ โ๐ณ ๐ ( ๐ + ๐ // ๐ณ + ๐ด ) , ๐ = โ . (3.5)The sum of all probabilities from every ๐ -pseudo-distribution (3.5) is ๐ ( ๐ // ๐ด ) = P ( ter ( ๐ // ๐ด )) , theprobability of a terraced event, generated by the frame s.e. ๐ด , in which the given s.e. ๐ณ ( โฉ ๐// ๐ด ) is in-serted.Thus, the only difference of e.p.d.โs of ๐ -inserteds.e.โs from their event-probabilistic ๐ -pseudo-distributions, lies in the fact that the sums of theprobabilities of the terraced events, from which the ๐ -pseudo-distributions are composed, are normal-ized not by unity, but by the probabilities of the cor-responding frame terraces events ๐ ( ๐ // ๐ด ) . And thesum of the normalizing constants by ๐ โ ๐ด is obvi-ously equal to one. Note 5 (symmetry of inserted and frame s.e.โs).
InDe๏ฌnition 6 the s.e. ๐ณ and ๐ด can always beswapped, i.e., to take the s.e. ๐ณ on a role of the frame one, and the s.e. ๐ด to take on a role of s.e.,that generates ๐ -inserted s.e.โs for every ๐ โ ๐ณ : ๐ด ( โฉ ๐// ๐ณ ) = ๐ด ( โฉ ) { ter ( ๐// ๐ณ ) } == { ๐ฆ โฉ ter ( ๐// ๐ณ ) , ๐ฆ โ ๐ด} . (3.6) Note 6 (M-sum of the all inserted s.e.โs).
The M-sum of the all ๐ -inserted s.e.โs ๐ณ ( โฉ ๐// ๐ด ) for ๐ โ ๐ด forms the given s.e. ๐ณ : ๐ณ = (๏ธโ๏ธ)๏ธ ๐ โ๐ด ๐ณ ( โฉ ๐// ๐ด ) == ๐ณ ( โฉโ // ๐ด ) (+) ... (+) ๐ณ ( โฉ๐ด // ๐ด ) โ โ |๐ด| . (3.7) Note 7 (charcterization of ๐ -inserted s.e.โs by con-ditional e.p.d.โs of the 1st kind). The e.p.d. of ๐ -inserted s.e. ๐ณ ( โฉ ๐// ๐ด ) with every ๐ โ ๐ด has a M-sum is a sum by Minkowski.
OROBYEV form for ๐ โ ๐ณ : ๐ (๏ธ ๐ ( โฉ ) { ter ( ๐ // ๐ด ) } // ๐ณ ( โฉ ๐// ๐ด ) )๏ธ == {๏ธ ๐ ( ๐ + ๐ // ๐ณ + ๐ด ) , โ ฬธ = ๐ โ ๐ณ ,๐ ( ๐ // ๐ณ + ๐ด ) + 1 โ ๐ ( ๐ // ๐ด ) , ๐ = โ , = {๏ธ ๐ ( ๐//
๐ณ |
๐ // ๐ด ) ๐ ( ๐ // ๐ด ) , โ ฬธ = ๐ โ ๐ณ , โ (1 โ ๐ ( โ // ๐ณ |
๐ // ๐ด )) ๐ ( ๐ // ๐ด ) , ๐ = โ (3.8)where for every ๐ โ ๐ด ๐ ( ๐//
๐ณ |
๐ // ๐ด ) = ๐ ( ๐ + ๐ // ๐ณ + ๐ด ) ๐ ( ๐ // ๐ด ) , (3.9)i.e., the probabilities of the 1st kind, forming for ๐ โ ๐ณ the ๐ -conditional e.p.d. of the 1st kind ofthe s.e. ๐ณ with respect to terraced event ter ( ๐ // ๐ด ) generated by the s.e. ๐ด .In other words, ๐ -inserted s.e. ๐ณ ( โฉ ๐// ๐ด ) for ๐ โ๐ด are characterized by formulas (3.8) and ๐ -conditional e.p.d.โs of the 1st kind of the s.e. ๐ณ withrespect to the terraced event ter ( ๐ // ๐ด ) , generatedby the s.e. ๐ด . Note 8 (mutual characterization of conditionale.p.d.โs of the 1st kind and pseudo-distributions ofinserted s.e.โs).
The connection between each ๐ -pseudo-distribution of the ๐ -inserted s.e. with thecorresponding ๐ -conditional e.p.d. looks simpler.It is su๏ฌcient for each ๏ฌxed ๐ โ ๐ด to normalizeall its probabilities of โinsertedโ terraced events bythe probability of a terraced event ๐ ( ๐ // ๐ด ) in orderto obtain corresponding to the ๐ -conditional prob-abilities regarding the fact that the correspondingframe terraces event ter ( ๐ // ๐ด ) happened. As aresult, we have the following obvious inversionformulas: ๐ ( ๐//
๐ณ |
๐ // ๐ด ) = ๐ ( ๐ ) ( ๐ + ๐ // ๐ณ + ๐ด ) ๐ ( ๐ // ๐ด ) ,๐ ( ๐ ) ( ๐ + ๐ // ๐ณ + ๐ด ) = ๐ ( ๐//
๐ณ |
๐ // ๐ด ) ๐ ( ๐ // ๐ด ) . (3.10) Note 9 (about the appropriateness of the conceptof inserted s.e.โs).
It would seem, why developa theory of inserted s.e.โs, pseudo-distributionsof which are simply characterized by conditionale.p.d.โs. 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-de๏ฌnedโconditional eventsโ. But until now it has notbeen possible to give a satisfactory de๏ฌnition ofthe โconditional eventโ, except for my impracticalde๏ฌnition in [7]. So, the inserted s.e.โs are a com-pletely satisfactory โsurrogateโ de๏ฌnition of thesets of โconditional eventsโ. Such that e.p.d.โs of inserted s.e.โs, although they do not coincide withthe desired conditional e.p.d.โs, but are associatedwith them by well-de๏ฌned mutual-inverse trans-formations. As a result, inserted s.e.โs play the roleof a convenient eventological tool for working withconditional e.p.d.โs of a one set of events regardingterrace events generated by another set of events.
Example 1 (two inserted s.e.โs in a frame monoplet).
Let in formulas (3.1) the s.e. ๐ณ is an arbitrary set,and the s.e. ๐ด = { ๐ฆ } is a frame monoplet of events,which have the joint e.p.d. in a form: { ๐ ( ๐ + ๐ // ๐ณ + { ๐ฆ } ) , ๐ โ ๐ณ , ๐ โ { ๐ฆ }} . (3.11)Then there is the { ๐ฆ } -inserted s.e. and the โ -inserted s.e.: ๐ณ ( โฉ{ ๐ฆ } // { ๐ฆ } ) = { ๐ฅ โฉ ๐ฆ, ๐ฅ โ ๐ณ } , ๐ณ ( โฉโ // { ๐ฆ } ) = { ๐ฅ โฉ ๐ฆ ๐ , ๐ฅ โ ๐ณ } . (3.12)These inserted s.e.โs are characterized for every oftwo subsets of the monoplet ๐ = { ๐ฆ } โ { ๐ฆ } and ๐ = โ โ { ๐ฆ } by formulas (3.3) and by two correspondinge.p.d.โs { ๐ ( ๐// ๐ณ + { ๐ฆ } ) } , ๐ โ ๐ณ } , { ๐ ( ๐ + { ๐ฆ } // ๐ณ + { ๐ฆ } ) } , ๐ โ ๐ณ } . (3.13)which by formulas (3.5) de๏ฌne two ๐ -pseudo-distributions for ๐ โ ๐ณ : ๐ ( { ๐ฆ } ) ( ๐ + { ๐ฆ } // ๐ณ + { ๐ฆ } ) == {๏ธ ๐ ( ๐ + { ๐ฆ } // ๐ณ + { ๐ฆ } ) , ๐ ฬธ = โ ,๐ ( { ๐ฆ } // ๐ณ + { ๐ฆ } ) โ ๐ ( { ๐ฆ } // { ๐ฆ } ) ,๐ = โ , = {๏ธ ๐ ( ๐ + { ๐ฆ } // ๐ณ + { ๐ฆ } ) , ๐ ฬธ = โ ,๐ ( { ๐ฆ } // ๐ณ + { ๐ฆ } ) โ ๐ ๐ฆ , ๐ = โ . (3.14) ๐ ( โ ) ( ๐// ๐ณ + { ๐ฆ } ) == {๏ธ ๐ ( ๐// ๐ณ + { ๐ฆ } ) , ๐ ฬธ = โ ,๐ ( โ // ๐ณ + { ๐ฆ } ) โ ๐ ( โ // { ๐ฆ } ) , ๐ = โ , = {๏ธ ๐ ( ๐// ๐ณ + { ๐ฆ } ) , ๐ ฬธ = โ ,๐ ( โ // ๐ณ + { ๐ฆ } ) โ ๐ ๐ฆ , ๐ = โ , (3.15)where ๐ ๐ฆ = P ( ๐ฆ ) is a probability of the frame event ๐ฆ โ { ๐ฆ } .First of all, note that the sum of the probabilitiesof terraced events from the { ๐ฆ } -pseudo-distribution(3.14) is ๐ ๐ฆ , and the probabilities of the โ -pseudo-distribution (3.15) is โ ๐ ๐ฆ ; and secondly, that thesetwo pseudo-distributions de๏ฌne a joint e.p.d. of thes.e. ๐ณ and the monoplet { ๐ฆ } , i.e., e.p.d. of the s.e. ๐ณ + { ๐ฆ } , which is related to them by fairly obviousformulas for ๐ โ ๐ณ + { ๐ฆ } : ๐ ( ๐// ๐ณ + { ๐ฆ } ) = {๏ธ ๐ ( { ๐ฆ } ) ( ๐// ๐ณ + { ๐ฆ } ) , ๐ฆ โ ๐,๐ ( โ ) ( ๐// ๐ณ + { ๐ฆ } ) , ๐ฆ ฬธโ ๐. (3.16) HE XIV FAMEMSโ2015 C
ONFERENCE
The formulas (3.16) are recurrent, connecting thee.p.d. of s.e. ๐ณ + { ๐ฆ } with two pseudo-distributionsof the inserted s.e. ๐ณ โฒ = ๐ณ ( โฉ{ ๐ฆ } // ๐ด ) and ๐ณ โฒโฒ = ๐ณ ( โฉโ // ๐ด ) whose power is less by one. The inversionformulas (3.10) allow recurrence formulas (3.16) toexpress the e.p.d. of ๐ณ + { ๐ฆ } via the conditionale.p.d. with respect to one of its events ๐ฆ โ ๐ณ + { ๐ฆ } and its complements ๐ฆ ๐ = โฆ โ ๐ฆ : ๐ ( ๐// ๐ณ + { ๐ฆ } ) == โงโชโชโชโจโชโชโชโฉ ๐ ( ๐//
๐ณ | { ๐ฆ } // { ๐ฆ } ) ๐ ๐ฆ , ๐ = ๐ + { ๐ฆ } ,๐ โ ๐ณ ; ๐ ( ๐//
๐ณ | โ // { ๐ฆ } )(1 โ ๐ ๐ฆ ) , ๐ = ๐,๐ โ ๐ณ , (3.17)where ๐ โ ๐ณ + { ๐ฆ } . Note that these formulas, like(3.16), can be used recursively to express the e.p.d.of s.e. ๐ณ + { ๐ฆ } through two conditional e.p.d.โs ofthe s.e. ๐ณ whose power is less by one. Definition 8 (inserted Kopulas).
The ๐ -Kopulasof ๐ -inserted ๐ -s.e.โs ๐ณ ( โฉ ๐// ๐ด ) = ๐ณ ( โฉ ) { ter ( ๐ // ๐ด ) } == { ๐ฅ โฉ ter ( ๐ // ๐ด ) , ๐ฅ โ ๐ณ } , (3.18)which for each ๐ โ ๐ด are de๏ฌned (see De๏ฌnition6) as intersections by Minkowsi of the s.e. ๐ณ withterraced events ter ( ๐ // ๐ด ) , generated by the s.e. ๐ด ,are called the ๐ -inserted ๐ -Kopulas with respect tothe s.e. ๐ด . Such ๐ -inserted ๐ -Kopulas character-izes e.p.d.โs of the 1st kind of ๐ -inserted ๐ -s.e.โs byformulas for ๐ โ ๐ณ ๐ ( ๐ ( โฉ ) { ter ( ๐ // ๐ด ) } ) == K ( ๐ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ ๐// ๐ด ) // ๐ณ ( โฉ ๐// ๐ด ) ) )๏ธ , (3.19)where ห ๐ ( ๐ |๐ณ ( โฉ ๐// ๐ด ) // ๐ณ ( โฉ ๐// ๐ด ) ) = ห ๐ ( ๐ ) == {๏ธ ๐ ( ๐ ) ๐ฅ , ๐ฅ โ ๐ณ }๏ธ = {๏ธ P ( ๐ฅ โฉ ter ( ๐ // ๐ด )) , ๐ฅ โ ๐ณ }๏ธ (3.20)is the set of probabilities of โinsertedโ marginalevents from the ๐ณ ( โฉ ๐// ๐ด ) , and ห ๐ ( ๐ | ๐ ( โฉ ๐// ๐ด ) // ๐ณ ( โฉ ๐// ๐ด ) ) = {๏ธ ๐ ( ๐ ) ๐ฅ , ๐ฅ โ ๐ }๏ธ ++ {๏ธ ๐ ( ๐ // ๐ณ ) โ ๐ ( ๐ ) ๐ฅ , ๐ฅ โ ๐ณ โ ๐ }๏ธ (3.21)are ๐ -phenomena of the set of โinsertedโ marginalprobabilities ห ๐ ( ๐ ) .We also need to de๏ฌne an inserted ๐ -pseudo-Kopula with respect to the s.e. ๐ด , which character-izes the ๐ -pseudo-distribution of the ๐ -inserted s.e. ๐ณ ( โฉ ๐// ๐ด ) , inserted into the terraces event ter ( ๐ // ๐ด ) generated by the s.e. ๐ด . Although the ๐ -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 ๐ -pseudo-Kopula of the ๐ -pseudo-distribution of ๐ -inserted s.e. ๐ณ ( โฉ ๐ ) = ๐ณ ( โฉ ๐// ๐ด ) = ๐ณ ( โฉ ) { ter ( ๐ // ๐ด ) } (3.22)with respect to the s.e. ๐ด is a such function K ( ๐ ) on ๐ณ -hypercube with sides [0 , ๐ ( ๐ // ๐ด )] that1) is non-negative : K ( ๐ ) (๏ธ ห ๐ค ( ๐ | ๐ ( โฉ ๐ ) // X ( โฉ ๐ ) ) )๏ธ (cid:62) (3.23)for ห ๐ค ( ๐ | ๐ ( โฉ ๐ ) // X ( โฉ ๐ ) ) โ [0 , ๐ ( ๐ // ๐ด )] โ๐ณ ๐ โ X ;2) satis๏ฌes the ๐ -marginal equalities for ๐ฅ โ X : โ๏ธ ๐ฅ โ ๐ โ X K ( ๐ ) (๏ธ ห ๐ค ( ๐ | ๐ ( โฉ ๐ ) // X ( โฉ ๐ ) ) )๏ธ = ๐ค ( ๐ | ๐ ( โฉ ๐ ) // X ( โฉ ๐ ) ) ๐ฅ โฉ ter ( ๐// ๐ณ ) (3.24)where ห ๐ค ( ๐ | ๐ ( โฉ ๐ ) // ๐ณ ( โฉ ๐ ) ) = {๏ธ ๐ค ( ๐ | ๐ ( โฉ ๐ ) // ๐ณ ( โฉ ๐ ) ) ๐ฅ โฉ ter ( ๐// ๐ณ ) , ๐ฅ โ ๐ณ }๏ธ (3.25)is a ๐ -phenomenon of the ๐ณ -set of marginal prob-abilities of the ๐ -pseudo-distribution of ๐ -inserteds.e. ๐ณ ( โฉ ๐ ) , i.e., ๐ค ( ๐ | ๐ ( โฉ ๐ ) // ๐ณ ( โฉ ๐ ) ) ๐ฅ โฉ ter ( ๐// ๐ณ ) == {๏ธ ๐ค ๐ฅ โฉ ter ( ๐// ๐ณ ) , ๐ฅ โ ๐,๐ ( ๐ // ๐ด ) โ ๐ค ๐ฅ โฉ ter ( ๐// ๐ณ ) , ๐ฅ โ ๐ณ โ ๐. (3.26)From (3.24) and (3.26) it follows the probabilistic ๐ -normalization of pseudo-Kopula : โ๏ธ ๐ โ๐ณ K ( ๐ ) (๏ธ ห ๐ค ( ๐ | ๐ ( โฉ ๐ ) // ๐ณ ( โฉ ๐ ) ) )๏ธ = ๐ ( ๐ // ๐ด ) . (3.27)And from (3.23) and (3.27) it follows the terrace-by-terrace probabilistic ๐ -normalization of pseudo-Kopula : (cid:54) K ( ๐ ) (๏ธ ห ๐ค ( ๐ | ๐ ( โฉ ๐ ) // ๐ณ ( โฉ ๐ ) ) )๏ธ (cid:54) ๐ ( ๐ // ๐ด ) (3.28)for ๐ โ X .Such ๐ -pseudo-Kopulas characterize the ๐ -pseudo-distribution (3.5) of ๐ -inserted s.e.โs ๐ณ ( โฉ ๐ ) by for-mulas for ๐ โ ๐ณ ๐ ( ๐ ) ( ๐ + ๐ // ๐ณ + ๐ด ) == K ( ๐ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ ๐// ๐ด ) // ๐ณ ( โฉ ๐// ๐ด ) ) )๏ธ , (3.29) OROBYEV where ห ๐ ( ๐ |๐ณ ( โฉ ๐// ๐ด ) // ๐ณ ( โฉ ๐// ๐ด ) ) = ห ๐ ( ๐ ) == {๏ธ ๐ ( ๐ ) ๐ฅ , ๐ฅ โ ๐ณ }๏ธ = {๏ธ P ( ๐ฅ โฉ ter ( ๐ // ๐ด )) , ๐ฅ โ ๐ณ }๏ธ (3.30)is a set of ๐ -marginal probabilities, coinciding withthe set of marginal probabilities of ๐ -inserted s.e.โs ๐ณ ( โฉ ๐ ) , and ห ๐ ( ๐ | ๐ ( โฉ ๐// ๐ด ) // ๐ณ ( โฉ ๐// ๐ด ) ) = {๏ธ ๐ ( ๐ ) ๐ฅ , ๐ฅ โ ๐ }๏ธ ++ {๏ธ ๐ ( ๐ // ๐ณ ) โ ๐ ( ๐ ) ๐ฅ , ๐ฅ โ ๐ณ โ ๐ }๏ธ (3.31)are ๐ -phenomena of the set ๐ -marginal probabili-ties ห ๐ ( ๐ ) . Definition 10 (conditional Kopulas).
The ๐ -Kopulas, characterizing ๐ -inserted e.p.d.โs of the1st kind of the ๐ -s.e. ๐ณ with respect to the terracedevent ter ( ๐ // ๐ด ) , generated by the s.e. ๐ด , i.e., e.p.d.โsof the 1st kind, de๏ฌned by joint e.p.d. ๐ณ and ๐ด byformulas with ๏ฌxed ๐ โ ๐ด for ๐ โ ๐ณ : ๐ ( ๐//
๐ณ |
๐ // ๐ด ) = ๐ ( ๐ + ๐ // ๐ณ + ๐ด ) ๐ ( ๐ // ๐ด ) , (3.32)are called the ๐ -conditional ๐ -Kopulas of the ๐ -s.e. ๐ณ with respect to the terraced event ter ( ๐ // ๐ด ) , gen-erated by the s.e. ๐ด .Such ๐ -cvonditional ๐ -Kopulas characterize the ๐ -conditional e.p.d. of the 1st kind (3.32) by formulasfor ๐ โ ๐ณ : ๐ ( ๐//
๐ณ |
๐ // ๐ด ) = K | ๐ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |
๐// ๐ด ) )๏ธ , (3.33)where ห ๐ ( ๐ |๐ณ // ๐ณ |
๐// ๐ด ) = ห ๐ | ๐ = {๏ธ ๐ | ๐๐ฅ , ๐ฅ โ ๐ณ }๏ธ == {๏ธ P ( ๐ฅ โฉ ter ( ๐ // ๐ด )) /๐ ( ๐ // ๐ด ) , ๐ฅ โ ๐ณ }๏ธ (3.34)is a set of conditional marginal probabilities ofevents ๐ฅ โ ๐ณ with respect to the terraced event ter ( ๐ // ๐ด ) , and ห ๐ ( ๐ | ๐//
๐ณ |
๐// ๐ด ) == {๏ธ ๐ | ๐๐ฅ , ๐ฅ โ ๐ }๏ธ + {๏ธ โ ๐ | ๐๐ฅ , ๐ฅ โ ๐ณ โ ๐ }๏ธ (3.35)are ๐ -phenomenon of the set of conditionalmarginal probabilities ห ๐ | ๐ . Note 10 (connection between conditional and โโin-sertedโโ marginal probaabilities).
Conditionalmarginal probabilities are connected with โin-sertedโ marginal probabilities for ๐ฅ โ ๐ณ by theformula of conditional probability: ๐ | ๐๐ฅ = 1 ๐ ( ๐ // ๐ด ) ๐ ( ๐ ) ๐ฅ , (3.36)since โinsertedโ marginal probabilities (3.20) areprobabilities of intersections of events ๐ฅ โ ๐ณ with the terraced event ter ( ๐ // ๐ด ) . The connection be-tween the corresponding set of conditional โin-sertedโ marginal probabilities we shall write in thesimilar way: ห ๐ | ๐ = 1 ๐ ( ๐ // ๐ด ) ห ๐ ( ๐ ) , ห ๐ ( ๐ | ๐//
๐ณ |
๐// ๐ด ) = 1 ๐ ( ๐ // ๐ด ) ห ๐ ( ๐ | ๐ ( โฉ ๐// ๐ด ) // ๐ณ ( โฉ ๐// ๐ด ) ) . (3.37) Note 11 (connection between conditional Kopulasand inserted pseudo-Kopulas).
From De๏ฌnition 10of conditional Kopula and De๏ฌnition 9 of insertedpseudo-Kopula with respect to the s.e. ๐ด , 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 ๐ณ for ๐ โ ๐ณ : K | ๐ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |
๐// ๐ด ) )๏ธ == 1 ๐ ( ๐ // ๐ด ) K ( ๐ ) (๏ธ ๐ ( ๐ // ๐ด )ห ๐ ( ๐ | ๐//
๐ณ |
๐// ๐ด ) )๏ธ , K ( ๐ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ ๐// ๐ด ) // ๐ณ ( โฉ ๐// ๐ด ) ) )๏ธ == ๐ ( ๐ // ๐ด ) K | ๐ (๏ธ ๐ ( ๐ // ๐ด ) ห ๐ ( ๐ | ๐ ( โฉ ๐// ๐ด ) // ๐ณ ( โฉ ๐// ๐ด ) ) )๏ธ . (3.38) Note 12 (two formulas of full probability for aKopula).
The Kopula K of s.e. ๐ณ is expressedthrough ๐ -conditional Kpulas K | ๐ for ๐ โ ๐ด by theusual formula of full probability: K (๏ธ ห ๐ ( ๐ | ๐// ๐ณ ) )๏ธ = โ๏ธ ๐ โ๐ด K | ๐ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |
๐// ๐ด ) )๏ธ ๐ ( ๐ // ๐ด ) . (3.39)From (3.39) and (3.38) we obtain an analogue of theformula of total probability โ the representationof the Kopula of s.e. ๐ณ in the form of sum of ๐ -pseudo-Kopulas by ๐ โ ๐ด : K (๏ธ ห ๐ ( ๐ | ๐// ๐ณ ) )๏ธ = โ๏ธ ๐ โ๐ด K ( ๐ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ ๐// ๐ด ) // ๐ณ ( โฉ ๐// ๐ด ) ) )๏ธ . (3.40) Note 13 (Kopula of a sum of sets).
A Kopula ofsum ๐ณ + ๐ด of two s.e.โs ๐ณ and ๐ด characterizes theirjoint e.p.d. of the 1st kind and by de๏ฌnition has theform ๐ ( ๐ + ๐ // ๐ณ + ๐ด ) = K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + ๐ด ) )๏ธ , (3.41)where ห ๐ ( ๐ | ๐ + ๐// ๐ณ + ๐ด ) = { ๐ ๐ฅ , ๐ฅ โ ๐ } + { ๐ ๐ฆ , ๐ฆ โ ๐ } ++ { โ ๐ ๐ฅ , ๐ฅ โ ๐ณ โ ๐ } + { โ ๐ ๐ฆ , ๐ฆ โ ๐ด โ ๐ } (3.42)is the ( ๐ + ๐ ) -phenomenon of the set of marginalprobabilities ห ๐ ( ๐ |๐ณ + ๐ด // ๐ณ + ๐ด ) = { ๐ ๐ฅ , ๐ฅ โ ๐ณ } + { ๐ ๐ฆ , ๐ฆ โ ๐ด} (3.43) HE XIV FAMEMSโ2015 C
ONFERENCE for the sum ๐ณ + ๐ด .From previous formulas (3.32), (3.33), and (3.29) fora inserted pseudo-Kopula and conditional Kopulawe obtain formulas K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + ๐ด ) )๏ธ == K ( ๐ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ ๐// ๐ด ) // ๐ณ ( โฉ ๐// ๐ด ) ) )๏ธ , (3.44) K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + ๐ด ) )๏ธ == K | ๐ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |
๐// ๐ด ) )๏ธ K (๏ธ ห ๐ ( ๐ | ๐// ๐ด ) )๏ธ , (3.45)that for each ๐ โ ๐ด connect the Kopula of sum ๐ณ + ๐ด with the product of ๐ -conditional Kopula ๐ณ with respect to ๐ด and the value of Kopula ๐ด at ๐ -phenomenon; and also with the ๐ -inserted pseudo-Kopula of ๐ณ which is inserted in the terraced event ter ( ๐ // ๐ด ) , generated by ๐ด . The basis of the frame method of constructing Kop-ula is a rather simple idea of composing an arbi-trary ๐ -s.e. X using the recurrence frame formula: X = { ๐ฅ , ๐ฅ , ..., ๐ฅ ๐ โ } = { ๐ฅ } + ๐ณ , (3.46)where ( ๐ โ -s.e.โs ๐ณ = X โ { ๐ฅ } = { ๐ฅ , ..., ๐ฅ ๐ โ } = (๏ธ ๐ณ โฒ (+) ๐ณ โฒโฒ )๏ธ (3.47)are composed from two ( ๐ โ -s.e.โs ๐ณ โฒ and ๐ณ โฒโฒ byset-theoretic operation of ๐ -union and de๏ฌned asthe inserted s.e.โs in the frame monoplet { ๐ฅ } by thefollowing formulas: ๐ณ โฒ = ๐ณ ( โฉ{ ๐ฅ } // { ๐ฅ } ) == { ๐ฅ } ( โฉ ) ๐ณ = { ๐ฅ โฉ ๐ฅ , ..., ๐ฅ โฉ ๐ฅ ๐ โ } , ๐ณ โฒโฒ = ๐ณ ( โฉโ // { ๐ฅ } ) == { ๐ฅ ๐ } ( โฉ ) ๐ณ = { ๐ฅ ๐ โฉ ๐ฅ , ..., ๐ฅ ๐ โฉ ๐ฅ ๐ โ } . (3.48)This simple idea allows us to ๏ฌnd the recurrentframe formulas for the ๐ -Kopula of s.e. X asfunctions of the set of marginal probabilities ห ๐ = { ๐ , ๐ , . . . , ๐ ๐ โ } .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 (ห ๐ ) = Recursion (๏ธ K ( { ๐ฅ } ) ๐ณ โฒ (ห ๐ ) , K ( โ ) ๐ณ โฒโฒ (ห ๐ ) )๏ธ , (3.49) K X (ห ๐ ) = Recursion (๏ธ K |{ ๐ฅ }๐ณ โฒ (ห ๐ ) , K |โ ๐ณ โฒโฒ (ห ๐ ) , ๐ )๏ธ , (3.50) -intersection and -union are an intersection and union odsets by Minkowski (see details in [1]). for the ๐ -Kopula of ๐ -s.e. X through known proba-bility ๐ of the event ๐ฅ and together with it throughtwo known inserted pseudo- ( ๐ โ -Kopulas (see Def-inition 9), i.e., through pseudo- ( ๐ โ -Kopulas of in-serted ( ๐ โ -s.e.โs ๐ณ โฒ and ๐ณ โฒโฒ in the frame mono-plet { ๐ฅ } , either through two known conditional ( ๐ โ -Kopulas (see De๏ฌnition 10) with respect tothe frame monoplet { ๐ฅ } of the same ๐ณ โฒ and ๐ณ โฒโฒ .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 = ๐ณ + { ๐ฅ } : K X (ห ๐ ) = K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + { ๐ฅ } ) )๏ธ , K ( { ๐ฅ } ) ๐ณ โฒ (ห ๐ ) = K ( { ๐ฅ } ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒ ) )๏ธ , K ( โ ) ๐ณ โฒโฒ (ห ๐ ) = K ( โ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) )๏ธ , K |{ ๐ฅ }๐ณ โฒ (ห ๐ ) = K |{ ๐ฅ } (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |{ ๐ฅ } // { ๐ฅ } ) )๏ธ , K |โ ๐ณ โฒโฒ (ห ๐ ) = K |โ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |โ // { ๐ฅ } ) )๏ธ . (3.51) Note 14 (about term โโframeโโ).
Although in formu-las (3.46) and (3.47) only monoplet { ๐ฅ } is a frame set, we shall call frame (with respect to this mono-plet) the s.e. X itself, construcyed from two inserteds.e.โs ๐ณ โฒ = ๐ณ ( โฉ{ ๐ฅ } // { ๐ฅ } ) and ๐ณ โฒโฒ = ๐ณ ( โฉโ // { ๐ฅ } ) , 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 (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + { ๐ฅ } ) )๏ธ == โงโชโจโชโฉ K ( { ๐ฅ } ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒ ) )๏ธ , ๐ = { ๐ฅ } , K ( โ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) )๏ธ , ๐ = โ , (3.52) K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + { ๐ฅ } ) )๏ธ == โงโชโจโชโฉ K |{ ๐ฅ } (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |{ ๐ฅ } // { ๐ฅ } ) )๏ธ ๐ , ๐ = { ๐ฅ } , K |โ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |โ // { ๐ฅ } ) )๏ธ (1 โ ๐ ) , ๐ = โ . (3.53)Although formulas (3.46) and (3.47) are satis๏ฌed 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.โs 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 ห ๐ = { ๐ , ๐ , ..., ๐ ๐ โ } are not more than half, forexample: / (cid:62) ๐ (cid:62) ๐ (cid:62) ... (cid:62) ๐ ๐ โ , (3.54)then the both inserted s.e.โs ๐ณ โฒ = { ๐ฅ โฒ , ..., ๐ฅ โฒ ๐ โ } and ๐ณ โฒโฒ = { ๐ฅ โฒโฒ , ..., ๐ฅ โฒโฒ ๐ โ } , and together with them and thes.e. ๐ณ are also half-rare by its De๏ฌnition (3.48). Inother words, their marginal probabilities from ห ๐ โฒ = { ๐ โฒ , ..., ๐ โฒ ๐ โ } ห ๐ โฒโฒ = { ๐ โฒโฒ , ..., ๐ โฒโฒ ๐ โ } do not exceed thecorresponding marginal probabilities events fromthe frame s.e. X : ๐ (cid:62) ๐ โฒ , ..., ๐ ๐ โ (cid:62) ๐ โฒ ๐ โ ,๐ (cid:62) ๐ โฒโฒ , ..., ๐ ๐ โ (cid:62) ๐ โฒโฒ ๐ โ , (3.55)and marginal probabilities from ห ๐ ๐ โ = { ๐ , ..., ๐ ๐ โ } are half-rare by de๏ฌnition. Thus,any half-rare ๐ -s.e. is composed by the framemethod with the formula (3.47) from two inserted ( ๐ โ -s.e.โs ๐ณ โฒ and ๐ณ โฒโฒ , which are required to behalf-rare. Lemma 4 (about independent half-rare s.e.โs , con-structed by the frame method from two inserted half-rares.e.โs ). That in the family of half-rare s.e.โs X withsets of marginal probabilitiues ห ๐ , constructed bythe frame method from two inserted half-rare s.e.โs ๐ณ โฒ and ๐ณ โฒโฒ , 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 = { ๐ฅ } + ๐ณ = { ๐ฅ } + ( ๐ณ โฒ (+) ๐ณ โฒโฒ ) by thefollowing way: ห ๐ โฒ = {๏ธ ๐ โฒ , . . . , ๐ โฒ ๐ โ }๏ธ = { ๐ ๐ , . . . , ๐ ๐ โ ๐ } , ห ๐ โฒโฒ = {๏ธ ๐ โฒโฒ , . . . , ๐ โฒโฒ ๐ โ }๏ธ = { ๐ (1 โ ๐ ) , . . . , ๐ ๐ โ (1 โ ๐ ) } ; (3.56)and su๏ฌcient so that the e.p.d. of the 1st kind ofinserted s.e.โs ๐ณ โฒ and ๐ณ โฒโฒ to be calculated from theformulas for ๐ โ ๐ณ : ๐ ( ๐ โฒ // ๐ณ โฒ ) = P (๏ธ โ๏ธ ๐ฅ โฒ โ ๐ โฒ ๐ฅ โฒ โ๏ธ ๐ฅ โฒ โ๐ณ โฒ โ ๐ โฒ ๐ฅ โฒ ๐ )๏ธ == โงโชโชโจโชโชโฉ ๐ โ๏ธ ๐ฅ โ ๐ ๐ ๐ฅ โ๏ธ ๐ฅ โ๐ณ โ ๐ (1 โ ๐ ๐ฅ ) , ๐ ฬธ = โ ,๐ โ๏ธ ๐ฅ โ๐ณ (1 โ ๐ ๐ฅ ) + 1 โ ๐ , ๐ = โ ,๐ ( ๐ โฒโฒ // ๐ณ โฒโฒ ) = P (๏ธ โ๏ธ ๐ฅ โฒโฒ โ ๐ โฒโฒ ๐ฅ โฒโฒ โ๏ธ ๐ฅ โฒโฒ โ๐ณ โฒโฒ โ ๐ โฒโฒ ๐ฅ โฒโฒ ๐ )๏ธ == โงโชโชโจโชโชโฉ (1 โ ๐ ) โ๏ธ ๐ฅ โ ๐ ๐ ๐ฅ โ๏ธ ๐ฅ โ๐ณ โ ๐ (1 โ ๐ ๐ฅ ) , ๐ ฬธ = โ , (1 โ ๐ ) โ๏ธ ๐ฅ โ๐ณ (1 โ ๐ ๐ฅ ) + ๐ , ๐ = โ , (3.57)where ๐ โฒ = { ๐ฅ โฒ , ๐ฅ โ ๐ } = { ๐ฅ โฉ ๐ฅ, ๐ฅ โ ๐ } โ ๐ณ โฒ ,๐ โฒโฒ = { ๐ฅ โฒโฒ , ๐ฅ โ ๐ } = { ๐ฅ ๐ โฉ ๐ฅ, ๐ฅ โ ๐ } โ ๐ณ โฒโฒ . (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 ๐ = 1 , ..., ๐ โ : ๐ โฒ ๐ = P ( ๐ฅ โฉ ๐ฅ ๐ ) = ๐ ๐ ๐ ,๐ โฒโฒ ๐ = P ( ๐ฅ ๐ โฉ ๐ฅ ๐ ) = ๐ ๐ (1 โ ๐ ) . (3.59)The s๏ฌciency 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.โs ๐ณ โฒ and ๐ณ โฒโฒ , which have the form for ๐ โ ๐ณ : ๐ ( ๐ + ๐ // ๐ณ + { ๐ฅ } ) == โงโชโชโชโชโชโชโจโชโชโชโชโชโชโฉ ๐ ( ๐ โฒ // ๐ณ โฒ ) , ๐ = { ๐ฅ } , ๐ ฬธ = โ ,๐ ( โ // ๐ณ โฒ ) โ ๐ , ๐ = { ๐ฅ } , ๐ = โ ,๐ ( ๐ โฒโฒ // ๐ณ โฒโฒ ) , ๐ = โ , ๐ ฬธ = โ ,๐ ( โ // ๐ณ โฒโฒ ) โ ๐ , ๐ = โ , ๐ = โ . (3.60)Demanding (3.60) to perform su๏ฌcient conditions(3.57), we get ๐ ( ๐ + ๐ // ๐ณ + { ๐ฅ } ) == โงโชโชโชโชโจโชโชโชโชโฉ ๐ โ๏ธ ๐ฅ โ ๐ ๐ ๐ฅ โ๏ธ ๐ฅ โ๐ณโ ๐ (1 โ ๐ ๐ฅ ) , ๐ = { ๐ฅ } , ๐ โ ๐ณ , (1 โ ๐ ) โ๏ธ ๐ฅ โ ๐ ๐ ๐ฅ โ๏ธ ๐ฅ โ๐ณโ ๐ (1 โ ๐ ๐ฅ ) ,๐ = โ , ๐ โ ๐ณ . (3.61)As a result, for the s.e. X we have the e.p.d. of the1st kind of independent events: ๐ ( ๐ + ๐ // ๐ณ + { ๐ฅ } ) == ๐ ( ๐// X ) = โ๏ธ ๐ฅ โ ๐ ๐ ๐ฅ โ๏ธ ๐ฅ โ X โ ๐ (1 โ ๐ ๐ฅ ) , (3.62)where ๐ = {๏ธ ๐ + { ๐ฅ } , ๐ = { ๐ฅ } , ๐ โ ๐ณ ,๐, ๐ = โ , ๐ โ ๐ณ . (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 de๏ฌnes the 1-Kopula of an ar-bitrary monoplet of events { ๐ฅ } with { ๐ฅ } -monopletof marginal probabilities ห ๐ = { ๐ ๐ฅ } โ [0 , ๐ฅ in theunique form: K (ห ๐ ) = K ( ๐ ๐ฅ ) = ๐ ๐ฅ , (4.1) By the way, the necessary condition also follows from (3.57). HE XIV FAMEMSโ2015 C
ONFERENCE which which provides 2 values on each ( ๐ | ห ๐ ) -phenomenon-dom by general formulas for ๐ โ{ ๐ฅ } : K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ } ) )๏ธ == {๏ธ K (1 โ ๐ ๐ฅ ) = 1 โ ๐ ๐ฅ , ๐ = โ , K ( ๐ ๐ฅ ) = ๐ ๐ฅ , ๐ = { ๐ฅ } . (4.2)However, the formula (4.2) can be generalized inthe following simple way: K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ } ) )๏ธ == {๏ธ K โ (1 โ ๐ ๐ฅ ) = 1 โ K { ๐ฅ } ( ๐ ๐ฅ ) , ๐ = โ , K { ๐ฅ } ( ๐ ๐ฅ ) , ๐ = { ๐ฅ } , (4.3)where K { ๐ฅ } is any function such that in half-rarevariables: K { ๐ฅ } : [0 , / โ [0 , / , (4.4)and in free variables: K { ๐ฅ } : [0 , โ [0 , . (4.5)In this case, the 1-Kopula (4.2) is an important spe-cial case of 1-Kopula (4.3) when K { ๐ฅ } ( ๐ ๐ฅ ) = ๐ ๐ฅ . 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 ห ๐ - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered frame half-rare doublet of events X = { ๐ฅ, ๐ฆ } = { ๐ฅ } + ๐ณ = { ๐ฅ } + (๏ธ ๐ณ โฒ (+) ๐ณ โฒโฒ )๏ธ (5.1)with the X -set of marginal probabilities ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ } ,where / (cid:62) ๐ ๐ฅ (cid:62) ๐ ๐ฆ (cid:62) , (5.2)letโs suppose that we have at our disposal two half-rare inserted monoplets of events ๐ณ โฒ = { ๐ฅ โฉ ๐ฆ } = { ๐ โฒ } ๐ณ โฒโฒ = { ๐ฅ ๐ โฉ ๐ฆ } = { ๐ โฒโฒ } , (5.3)with known 1-Kopulas: K โฒ (๏ธ ห ๐ ( ๐ | ๐// { ๐ โฒ } ) )๏ธ == {๏ธ K โฒ (1 โ ๐ ๐ โฒ ) = 1 โ ๐ ๐ โฒ , ๐ = โ , K โฒ ( ๐ ๐ โฒ ) = ๐ ๐ โฒ , ๐ = { ๐ โฒ } , (5.4) K โฒโฒ (๏ธ ห ๐ ( ๐ | ๐// { ๐ โฒโฒ } ) )๏ธ == {๏ธ K โฒโฒ (1 โ ๐ ๐ โฒโฒ ) = 1 โ ๐ ๐ โฒโฒ , ๐ = โ , K โฒโฒ ( ๐ ๐ โฒโฒ ) = ๐ ๐ โฒโฒ , ๐ = { ๐ โฒโฒ } , (5.5)By De๏ฌnition of inserted monoplets (5.3) (see Fig.7) ๐ โฒ = ๐ฅ โฉ ๐ฆ โ ๐ฅ,๐ โฒโฒ = ๐ฅ ๐ โฉ ๐ฆ โ ๐ฅ ๐ , (5.6)and also because of the ห ๐ -ordering assumption(5.2), we get that ๐ ๐ โฒ + ๐ ๐ โฒโฒ = ๐ ๐ฆ (cid:54) ๐ ๐ฅ (cid:54) / (cid:54) โ ๐ ๐ฅ . (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: ๐ ๐ โฒ + ๐ ๐ โฒโฒ = ๐ ๐ฆ , (5.8)and depend on only one parameter: ๐ ๐ โฒ โ [0 , ๐ ๐ฆ ] . (5.9) ฮฉ ๐ฅ๐ฆ ๐ โฒ ๐ โฉ ๐ฅ = ๐ฆ ๐ โฉ ๐ฅ ๐ โฒ = ๐ฅ โฉ ๐ฆ ๐ โฒโฒ = ๐ฅ ๐ โฉ ๐ฆ๐ โฒโฒ ๐ โฉ ๐ฅ ๐ = ๐ฆ ๐ โฉ ๐ฅ ๐ ฮฉ ๐ โฒ ๐ โฉ ๐ฅ = ๐ฆ ๐ โฉ ๐ฅ ๐ โฒ = ๐ฅ โฉ ๐ฆ ๐ โฒโฒ = ๐ฅ ๐ โฉ ๐ฆ๐ โฒโฒ ๐ โฉ ๐ฅ ๐ = ๐ฆ ๐ โฉ ๐ฅ ๐ โ โ ๐ฅ โ โ ๐ฅ ๐ Figure 7: Venn diagrams of the frame half-rare doublet of events X = { ๐ฅ, ๐ฆ } , / (cid:62) ๐ ๐ฅ (cid:62) ๐ ๐ฆ (up), and two inserted monoplets ๐ณ โฒ = { ๐ โฒ } and ๐ณ โฒโฒ = { ๐ โฒโฒ } (down) agreed with the frame doublet X in the followingsense: ๐ฆ = ๐ โฒ + ๐ โฒโฒ ๐ โฒ โ ๐ฅ, ๐ โฒโฒ โ ๐ฅ ๐ . OROBYEV We get the following formulas: ๐ ( ๐ฅ๐ฆ// { ๐ฅ, ๐ฆ } ) = ๐ ( ๐ โฒ // ๐ณ โฒ ) = ๐ ๐ โฒ ,๐ ( ๐ฅ// { ๐ฅ, ๐ฆ } ) = ๐ ( โ // ๐ณ โฒ ) โ ๐ ๐ฅ = ๐ ๐ฅ โ ๐ ๐ โฒ ,๐ ( ๐ฆ// { ๐ฅ, ๐ฆ } ) = ๐ ( ๐ โฒโฒ // ๐ณ โฒโฒ ) = ๐ ๐ฆ โ ๐ ๐ โฒ ,๐ ( โ // { ๐ฅ, ๐ฆ } ) = ๐ ( โ // ๐ณ โฒโฒ ) โ ๐ ๐ฅ = 1 โ ๐ ๐ฆ โ ๐ ๐ฅ + ๐ ๐ โฒ . (5.10)These formulas express the e.p.d. of the 1st kindof the ห ๐ -ordered half-rare frame doublet of events X = { ๐ฅ, ๐ฆ } through the e.p.d. of the 1st kindof inserted monoplets ๐ณ โฒ and ๐ณ โฒโฒ , and the prob-ability of frame event ๐ฅ , and, in the ๏ฌnal result,through their marginal probabilities ๐ ๐ฅ and ๐ ๐ฆ , andmarginal probability ๐ ๐ โฒ of the inserted monoplet ๐ณ โฒ = { ๐ โฒ } = { ๐ฅ โฉ ๐ฆ } .The formulas (5.10) express values of the 2-Kopulaof ห ๐ -ordered doublet X = { ๐ฅ, ๐ฆ } through 1-Kopulasof inserted monoplets ๐ณ โฒ = { ๐ โฒ } and ๐ณ โฒโฒ = { ๐ โฒโฒ } .Rewrite this in a form of an explicit recurrent for-mula: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K X (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโชโจโชโชโชโชโฉ K ๐ณ โฒ ( ๐ ๐ โฒ ) , ๐ = { ๐ฅ, ๐ฆ } , K ๐ณ โฒ (1 โ ๐ ๐ โฒ ) โ ๐ ๐ฅ , ๐ = { ๐ฅ } , K ๐ณ โฒโฒ ( ๐ ๐ โฒโฒ ) , ๐ = { ๐ฆ } , K ๐ณ โฒโฒ (1 โ ๐ ๐ โฒโฒ ) โ ๐ ๐ฅ , ๐ = โ . (5.11)Considering (2.34) and (5.9), we will continue: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K X (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโชโจโชโชโชโชโฉ ๐ ๐ โฒ , ๐ = { ๐ฅ, ๐ฆ } ,๐ ๐ฅ โ ๐ ๐ โฒ , ๐ = { ๐ฅ } ,๐ ๐ โฒโฒ , ๐ = { ๐ฆ } , โ ๐ ๐ฅ โ ๐ ๐ โฒโฒ , ๐ = โ , (5.12) = โงโชโชโชโชโจโชโชโชโชโฉ ๐ ๐ โฒ , ๐ = { ๐ฅ, ๐ฆ } ,๐ ๐ฅ โ ๐ ๐ โฒ , ๐ = { ๐ฅ } ,๐ ๐ฆ โ ๐ ๐ โฒ , ๐ = { ๐ฆ } , โ ๐ ๐ฅ โ ๐ ๐ฆ + ๐ ๐ โฒ , ๐ = โ . (5.13)We note, by the way, that the restriction (5.9) bythe assumption of ห ๐ -ordered (5.2) is a special caseof Frยดechet-inequalities: (cid:54) ๐ ๐ โฒ (cid:54) ๐ + ๐ฅ๐ฆ = min { ๐ ๐ฅ , ๐ ๐ฆ } = ๐ ๐ฆ . (5.14) Note 17 (frame mathod for otherwise ห ๐ -ordered half-rare doublet of events). For otherwise ห ๐ -ordered half-rare doublet of events X = { ๐ฆ, ๐ฅ } = { ๐ฆ } + ๐ณ = { ๐ฆ } + (๏ธ ๐ณ โฒ (+) ๐ณ โฒโฒ )๏ธ , (5.15)where ๐ณ โฒ = { ๐ฆ โฉ ๐ฅ } = { ๐ โฒ } ๐ณ โฒโฒ = { ๐ฆ ๐ โฉ ๐ฅ } = { ๐ โฒโฒ } , (5.16) ฮฉ ๐ฆ๐ฅ ๐ โฒ ๐ โฉ ๐ฆ = ๐ฅ ๐ โฉ ๐ฆ ๐ โฒ = ๐ฆ โฉ ๐ฅ ๐ โฒโฒ = ๐ฆ ๐ โฉ ๐ฅ๐ โฒโฒ ๐ โฉ ๐ฆ ๐ = ๐ฅ ๐ โฉ ๐ฆ ๐ ฮฉ ๐ โฒ ๐ โฉ ๐ฆ = ๐ฅ ๐ โฉ ๐ฆ ๐ โฒ = ๐ฆ โฉ ๐ฅ ๐ โฒโฒ = ๐ฆ ๐ โฉ ๐ฅ๐ โฒโฒ ๐ โฉ ๐ฆ ๐ = ๐ฅ ๐ โฉ ๐ฆ ๐ โ โ ๐ฆ โ โ ๐ฆ ๐ Figure 8: Venn diagrams of the frame, otherwise ordered half-rare dou-blet events X = { ๐ฆ, ๐ฅ } , / (cid:62) ๐ ๐ฆ (cid:62) ๐ ๐ฅ (up), and two inserted monoplets ๐ณ โฒ = { ๐ โฒ } and ๐ณ โฒโฒ = { ๐ โฒโฒ } (down), agreed with the frame doublet X inthe following sense: ๐ฅ = ๐ โฒ + ๐ โฒโฒ ๐ โฒ โ ๐ฆ, ๐ โฒโฒ โ ๐ฆ ๐ . with the X -set of marginal probabilities ห ๐ = { ๐ ๐ฆ , ๐ ๐ฅ } , / (cid:62) ๐ ๐ฆ (cid:62) ๐ ๐ฅ , the assumptions (5.18) take symmet-rical form: ๐ โฒ = ๐ฆ โฉ ๐ฅ โ ๐ฆ,๐ โฒโฒ = ๐ฆ ๐ โฉ ๐ฅ โ ๐ฆ ๐ . (5.17)By De๏ฌnition of inserted monoplets (5.16) (. . 8) ๐ โฒ = ๐ฅ โฉ ๐ฆ โ ๐ฆ,๐ โฒโฒ = ๐ฅ โฉ ๐ฆ ๐ โ ๐ฆ ๐ , (5.18)and also because of the assumption and also be-cause of the assumption of another ห ๐ -ordering, weget that ๐ ๐ โฒ + ๐ ๐ โฒโฒ = ๐ ๐ฅ (cid:54) ๐ ๐ฆ (cid:54) / (cid:54) โ ๐ ๐ฆ . (5.19)Consequently, 1-copulas of inserted monoplanesare connected by a restriction on the sum of theirmarginal probabilities: ๐ ๐ โฒ + ๐ ๐ โฒโฒ = ๐ ๐ฅ , (5.20)and depend on only one parameter: ๐ ๐ โฒ โ [0 , ๐ ๐ฅ ] . (5.21)By the assumptions, the following formulas are HE XIV FAMEMSโ2015 C
ONFERENCE valid: ๐ ( ๐ฅ๐ฆ// { ๐ฆ, ๐ฅ } ) = ๐ ( ๐ โฒ // ๐ณ โฒ ) = ๐ ๐ โฒ ,๐ ( ๐ฆ// { ๐ฆ, ๐ฅ } ) = ๐ ( โ // ๐ณ โฒ ) โ ๐ ๐ฆ = ๐ ๐ฆ โ ๐ ๐ โฒ ,๐ ( ๐ฅ// { ๐ฆ, ๐ฅ } ) = ๐ ( ๐ โฒโฒ // ๐ณ โฒโฒ ) = ๐ ๐ฅ โ ๐ ๐ โฒ ,๐ ( โ // { ๐ฆ, ๐ฅ } ) = ๐ ( โ // ๐ณ โฒโฒ ) โ ๐ ๐ฆ = 1 โ ๐ ๐ฅ โ ๐ ๐ฆ โ ๐ ๐ โฒ . (5.22)These formulas express the e.p.d. of the 1stkind otherwise ห ๐ -ordered half-rare frame doubletof events X through the e.p.d. of the 1st kind of inserted monoplets ๐ณ โฒ and ๐ณ โฒโฒ , and the probabilityof frame event ๐ฆ , and, in the ๏ฌnal result, throughown marginal probabilities ๐ ๐ฅ ๐ ๐ฆ , and the marginalprobabilities of inserted monoplet ๐ณ โฒ = { ๐ โฒ } = { ๐ฅ โฉ ๐ฆ } (see Fig. 8).The formulas (5.22) express values of the 2-Kopulaof ห ๐ -ordered doublet X = { ๐ฆ, ๐ฅ } through 1-Kopulasof inserted monoplets ๐ณ โฒ = { ๐ โฒ } and ๐ณ โฒโฒ = { ๐ โฒโฒ } .Rewrite this in the form of explicit recurrent for-mula: ๐ ( ๐// { ๐ฆ, ๐ฅ } ) = K X (๏ธ ห ๐ ( ๐ | ๐// { ๐ฆ,๐ฅ } ) )๏ธ == โงโชโชโชโจโชโชโชโฉ K ๐ณ โฒ ( ๐ ๐ โฒ ) , ๐ = { ๐ฅ, ๐ฆ } , K ๐ณ โฒ (1 โ ๐ ๐ โฒ ) โ ๐ ๐ฆ , ๐ = { ๐ฆ } , K ๐ณ โฒโฒ ( ๐ ๐ โฒโฒ ) , ๐ = { ๐ฅ } , K ๐ณ โฒโฒ (1 โ ๐ ๐ โฒโฒ ) โ ๐ ๐ฆ , ๐ = โ . (5.23)Continue: ๐ ( ๐// { ๐ฅ, ๐ฆ } ) = K X (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ == โงโชโชโชโจโชโชโชโฉ ๐ ๐ โฒ , ๐ = { ๐ฅ, ๐ฆ } ,๐ ๐ฆ โ ๐ ๐ โฒ , ๐ = { ๐ฆ } ,๐ ๐ โฒโฒ , ๐ = { ๐ฅ } , โ ๐ ๐ฅ โ ๐ ๐ โฒโฒ , ๐ = โ , (5.24) = โงโชโชโชโจโชโชโชโฉ ๐ ๐ โฒ , ๐ = { ๐ฅ, ๐ฆ } ,๐ ๐ฆ โ ๐ ๐ โฒ , ๐ = { ๐ฆ } ,๐ ๐ฅ โ ๐ ๐ โฒ , ๐ = { ๐ฅ } , โ ๐ ๐ฅ โ ๐ ๐ฆ + ๐ ๐ โฒ , ๐ = โ . (5.25)We note, as above, that the restriction (5.21) by theassumption of another ห ๐ -ordering is a special caseof Frยดechet inequalities: (cid:54) ๐ ๐ โฒ (cid:54) ๐ + ๐ฅ๐ฆ = min { ๐ ๐ฅ , ๐ ๐ฆ } = ๐ ๐ฅ . (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 = { ๐ฅ, ๐ฆ } = { ๐ฅ } + { ๐ฆ } = { ๐ฅ } + ๐ณ : K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + { ๐ฅ } ) )๏ธ == โงโชโจโชโฉ K ( { ๐ฅ } ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒ ) )๏ธ , ๐ = { ๐ฅ } , K ( โ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) )๏ธ , ๐ = โ , (5.27) K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + { ๐ฅ } ) )๏ธ == โงโชโจโชโฉ K |{ ๐ฅ } (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |{ ๐ฅ } // { ๐ฅ } ) )๏ธ ๐ ๐ฅ , ๐ = { ๐ฅ } , K |โ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |โ // { ๐ฅ } ) )๏ธ (1 โ ๐ ๐ฅ ) , ๐ = โ . (5.28)In the formulas (5.27) the pseudo-Kopulas K ( { ๐ฅ } ) and K ( โ ) of inserted monoplets ๐ณ โฒ and ๐ณ โฒโฒ , corre-spondingly, are de๏ฌned by the ๏ฌrst and the secondpairs of probabilities from (5.13) correspondingly,i.e., by formulas: K ( { ๐ฅ } ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒ ) )๏ธ == {๏ธ ๐ ๐ โฒ , ๐ = { ๐ฅ, ๐ฆ } ,๐ ๐ฅ โ ๐ ๐ โฒ , ๐ = { ๐ฅ } , (5.29) K ( โ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) )๏ธ == {๏ธ ๐ ๐ฆ โ ๐ ๐ โฒ , ๐ = { ๐ฆ } , โ ๐ ๐ฅ โ ๐ ๐ฆ + ๐ ๐ โฒ , ๐ = โ , (5.30)where, for example, for ๐ = { ๐ฆ }{ ๐ฆ } ( โฉ{ ๐ฅ } // { ๐ฅ } ) = { ๐ฆ โฉ ๐ฅ } โ ๐ณ โฒ = { ๐ โฒ } , { ๐ฆ } ( โฉโ // { ๐ฅ } ) = { ๐ฆ โฉ ๐ฅ ๐ } โ ๐ณ โฒโฒ = { ๐ฆ โ ๐ โฒ } , (5.31)and the corresponding sets of marginal probabili-ties of inserted monoplets ๐ณ โฒ and ๐ณ โฒโฒ have the form ห ๐ ( ๐ |{ ๐ฆ } ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒ ) = { ๐ ๐ โฒ } , ห ๐ ( ๐ |{ ๐ฆ } ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) = { ๐ ๐ฆ โ ๐ ๐ โฒ } . (5.32)In the formulas (5.28) the conditional Kopulas K |{ ๐ฅ } and K |โ are de๏ฌned by the ๏ฌrst and the secondpairs of probabilities from (5.13), normalized by ๐ ๐ฅ and by โ ๐ ๐ฅ correspondingly, i.e. by the formulas: K |{ ๐ฅ } (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |{ ๐ฅ } // { ๐ฅ } ) )๏ธ == {๏ธ ๐ ๐ฅ ๐ ๐ โฒ , ๐ = { ๐ฅ, ๐ฆ } , ๐ ๐ฅ ( ๐ ๐ฅ โ ๐ ๐ โฒ ) , ๐ = { ๐ฅ } , (5.33) K |โ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |โ // { ๐ฅ } ) )๏ธ == {๏ธ โ ๐ ๐ฅ ( ๐ ๐ฆ โ ๐ ๐ โฒ ) , ๐ = { ๐ฆ } , โ ๐ ๐ฅ (1 โ ๐ ๐ฅ โ ๐ ๐ฆ + ๐ ๐ โฒ ) , ๐ = โ . (5.34)The corresponding sets of marginal conditionalprobabilities of events ๐ฆ โ ๐ณ with respect tothe frame terraced events ter ( { ๐ฅ } // { ๐ฅ } ) = ๐ฅ ter ( โ // { ๐ฅ } ) = ๐ฅ ๐ correspondingly have the form: ห ๐ ( ๐ |๐ณ // ๐ณ |{ ๐ฅ } // { ๐ฅ } ) = {๏ธ ๐ ๐ โฒ ๐ ๐ฅ }๏ธ , ห ๐ ( ๐ |๐ณ // ๐ณ |โ // { ๐ฅ } ) = {๏ธ ๐ ๐ฆ โ ๐ ๐ โฒ โ ๐ ๐ฅ }๏ธ . (5.35) OROBYEV Remind, that Frยดechet restrictions on the functionalparameter ๐ ๐ โฒ = ๐ ๐ โฒ ( ๐ ๐ฅ , ๐ ๐ฆ ) for ห ๐ -ordered half-raredoublet of events X = { ๐ฅ, ๐ฆ } have the form: (cid:54) ๐ ๐ โฒ (cid:54) ๐ + ๐ฅ๐ฆ = min { ๐ ๐ฅ , ๐ ๐ฆ } = ๐ ๐ฆ , (5.36)and for otherwise ห ๐ -ordered half-rare doublet ofevents X = { ๐ฆ, ๐ฅ } have the form: (cid:54) ๐ ๐ โฒ (cid:54) ๐ + ๐ฅ๐ฆ = min { ๐ ๐ฅ , ๐ ๐ฆ } = ๐ ๐ฅ . (5.37) -Kopula First, without the frame method, which is not re-quired here, consider the simplest example of a -Kopula K โ ฮจ X of the ( ๐ โ -set of events X = { ๐ฅ, ๐ฆ, ๐ง } , i.e., a 1-function on the unit X -cube. In other words, construct such a nonnegativebounded numerical function K : [0 , โ X โ [0 , , that for all ๐ง โ X โ๏ธ ๐ฅ โ ๐ โ X K (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = ๐ค ๐ฅ . Such a simple example of a 1-function on X -cube isso-called independent ( ๐ โ -Kopula , which for allfree variables ห ๐ค = { ๐ค ๐ฅ , ๐ค ๐ฆ , ๐ค ๐ง } โ [0 , ๐ฅ โ [0 , ๐ฆ โ [0 , ๐ง = [0 , โ X is de๏ฌned by the formula: K ( ห ๐ค ) = ๐ค ๐ฅ ๐ค ๐ฆ ๐ค ๐ง , (6.1)that provides it on each ( ๐ | ห ๐ค ) -phenomenon-domthe following values: K (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = โ๏ธ ๐ฅ โ ๐ ๐ค ๐ฅ โ๏ธ ๐ฅ โ X โ ๐ (1 โ ๐ค ๐ฅ ) (6.2)for ๐ โ X . Indeed, as in the case of the doubletof events this function is a 1-function, since for all ๐ฅ โ X โ๏ธ ๐ฅ โ ๐ โ X (๏ธ โ๏ธ ๐ง โ ๐ ๐ค ๐ง โ๏ธ ๐ง โ X โ ๐ (1 โ ๐ค ๐ง ) )๏ธ = ๐ค ๐ฅ . The e.p.d. of the 1st kind of independent triplet ofevents X with the X -set of probabilities of events ห ๐ is de๏ฌned by values of the independent 3-Kopula(6.1) on ( ๐ | ห ๐ ) -phenomenon-dom by the general for-mulas of half-rare variables, i.e., for ๐ โ { ๐ฅ, ๐ฆ } : ๐ ( ๐// X ) = K (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ = โ๏ธ ๐ฅ โ ๐ ๐ ๐ฅ โ๏ธ ๐ฅ โ X โ ๐ (1 โ ๐ ๐ฅ ) . (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 = { ๐ฅ, ๐ฆ, ๐ง } , de๏ฌned onthe cube [0 , , in projections on planes, which areorthogonal to the axis ๐ ๐ฆ . Figure 9: The visualization of projections of the same three-dimensionalmap of Cartesian representation of independent 3-Kopula of the triplet X = { ๐ฅ, ๐ฆ, ๐ง } on the unit cube in conditional colors with values ofmarginal probaboility ๐ ๐ฆ = 0 . , ..., . , . , where the white color cor-responds to points in which probabilities of all terraced events are 1/8.The orientation of axes: ( ๐ ๐ฅ , ๐ ๐ง ) = (horizontal, vertical). In order to construct by the frame method the ห ๐ - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered frame half-rare triplet of events X = { ๐ฅ, ๐ฆ, ๐ง } with the X -set of marginal probabilities ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง } , where / (cid:62) ๐ ๐ฅ (cid:62) ๐ ๐ฆ (cid:62) ๐ ๐ง (cid:62) , (6.4)letโs suppose that X = { ๐ฅ } + { ๐ฆ, ๐ง } = { ๐ฅ } + ( ๐ณ โฒ (+) ๐ณ โฒโฒ ) (6.5)and in our disposal we have two inserted half-raredoublets of events ๐ณ โฒ = { ๐ โฒ , ๐ก โฒ } ๐ณ โฒโฒ = { ๐ โฒโฒ , ๐ก โฒโฒ } , with the known 2-Kopulas (see Fig. 10) which bythe de๏ฌnition satisfy the following inclusions: ๐ โฒ = ๐ฅ โฉ ๐ฆ โ ๐ฅ, ๐ก โฒ = ๐ฅ โฉ ๐ง โ ๐ฅ, ๐ โฒ โช ๐ก โฒ โ ๐ฅ,๐ โฒโฒ = ๐ฅ ๐ โฉ ๐ฆ โ ๐ฅ ๐ , ๐ก โฒโฒ = ๐ฅ ๐ โฉ ๐ง โ ๐ฅ ๐ , ๐ โฒโฒ โช ๐ก โฒโฒ โ ๐ฅ ๐ . (6.6)In view of the assumptions made (see Fig. 10) ter ( ๐ฅ๐ฆ๐ง// X ) = ๐ฅ โฉ ๐ฆ โฉ ๐ง = ๐ โฒ โฉ ๐ก โฒ = ter ( ๐ โฒ ๐ก โฒ // ๐ณ โฒ ) , ter ( ๐ฅ๐ฆ// X ) = ๐ฅ โฉ ๐ฆ โฉ ๐ง ๐ = ๐ โฒ โฉ ๐ก โฒ ๐ = ter ( ๐ โฒ // ๐ณ โฒ ) , ter ( ๐ฅ๐ง// X ) = ๐ฅ โฉ ๐ฆ ๐ โฉ ๐ง = ๐ โฒ ๐ โฉ ๐ก โฒ = ter ( ๐ก โฒ // ๐ณ โฒ ) , ter ( ๐ฆ๐ง// X ) = ๐ฅ ๐ โฉ ๐ฆ โฉ ๐ง = ๐ โฒโฒ โฉ ๐ก โฒโฒ = ter ( ๐ โฒโฒ ๐ก โฒโฒ // ๐ณ โฒโฒ ) , ter ( ๐ฆ// X ) = ๐ฅ ๐ โฉ ๐ฆ โฉ ๐ง ๐ = ๐ โฒโฒ โฉ ๐ก โฒโฒ ๐ = ter ( ๐ โฒโฒ // ๐ณ โฒโฒ ) , ter ( ๐ง// X ) = ๐ฅ ๐ โฉ ๐ฆ ๐ โฉ ๐ง = ๐ โฒโฒ ๐ โฉ ๐ก โฒโฒ = ter ( ๐ก โฒโฒ // ๐ณ โฒโฒ ) , (6.7)these 6 terraced events are de๏ฌned. All of them aregenerated by the frame half-rare triplet X , with the
00 T HE XIV FAMEMSโ2015 C
ONFERENCE ฮฉ ๐ฅ ๐ โฉ ๐ โฒโฒ ๐ โฉ ๐ก โฒโฒ ๐ ๐ฆ๐ฅ ๐ง ๐ โฒโฒ โฉ ๐ก โฒโฒ ๐ ๐ โฒ โฉ ๐ก โฒ ๐ ๐ฅ โฉ ๐ โฒ ๐ โฉ ๐ก โฒ ๐ ๐ โฒ โฉ ๐ก โฒ ๐ โฒโฒ โฉ ๐ก โฒโฒ ๐ โฒ ๐ โฉ ๐ก โฒ ๐ โฒโฒ ๐ โฉ ๐ก โฒโฒ ฮฉ ๐ฅ โฉ ๐ โฒ ๐ โฉ ๐ก โฒ ๐ ๐ฅ ๐ โฉ ๐ โฒโฒ ๐ โฉ ๐ก โฒโฒ ๐ ๐ โฒ โฉ ๐ก โฒ ๐ ๐ โฒโฒ โฉ ๐ก โฒโฒ ๐ ๐ โฒ โฉ ๐ก โฒ ๐ โฒโฒ โฉ ๐ก โฒโฒ ๐ โฒ ๐ โฉ ๐ก โฒ ๐ โฒโฒ ๐ โฉ ๐ก โฒโฒ ๐ โฒ ๐ โฒโฒ ๐ก โฒ ๐ก โฒโฒ โ โ ๐ฅ โ โ ๐ฅ ๐ Figure 10: Venn diagrams of the frame half-rare trip0let of events X = { ๐ฅ, ๐ฆ, ๐ง } , / (cid:62) ๐ ๐ฅ (cid:62) ๐ ๐ฆ (cid:62) ๐ ๐ง (up), and two inserted doublets ๐ณ โฒ = { ๐ โฒ , ๐ก โฒ } and ๐ณ โฒโฒ = { ๐ โฒโฒ , ๐ก โฒโฒ } (down), agreed with the frame triplet X in thefollowing sense: ๐ฆ = ๐ โฒ + ๐ โฒโฒ , ๐ง = ๐ก โฒ + ๐ก โฒโฒ ๐ โฒ โช ๐ก โฒ โ ๐ฅ, ๐ โฒโฒ โช ๐ก โฒโฒ โ ๐ฅ ๐ . exception of two terraced events ter ( ๐ฅ// X ) = ๐ฅ โฉ ๐ฆ ๐ โฉ ๐ง ๐ , ter ( โ // X ) = ๐ฅ ๐ โฉ ๐ฆ ๐ โฉ ๐ง ๐ , (6.8)that are de๏ฌned by the formulas: ter ( ๐ฅ// X ) = ๐ฅ โ ๐ โฒ โช ๐ก โฒ , ter ( โ // X ) = ๐ฅ ๐ โ ๐ โฒโฒ โช ๐ก โฒโฒ , (6.9)or, equivalently, ter ( ๐ฅ// X ) = ter ( โ // ๐ณ โฒ ) โ ๐ฅ ๐ , ter ( โ // X ) = ter ( โ // ๐ณ โฒโฒ ) โ ๐ฅ. (6.10)In view of this, we obtain the formulas: ๐ ( ๐ฅ๐ฆ๐ง// X ) = ๐ ( ๐ โฒ ๐ก โฒ // ๐ณ โฒ ) ,๐ ( ๐ฅ๐ฆ// X ) = ๐ ( ๐ โฒ // ๐ณ โฒ ) ,๐ ( ๐ฅ๐ง// X ) = ๐ ( ๐ก โฒ // ๐ณ โฒ ) ,๐ ( ๐ฅ// X ) = ๐ ( โ // ๐ณ โฒ ) โ ๐ ๐ฅ ,๐ ( ๐ฆ๐ง// X ) = ๐ ( ๐ โฒโฒ ๐ก โฒโฒ // ๐ณ โฒโฒ ) ,๐ ( ๐ฆ// X ) = ๐ ( ๐ โฒโฒ // ๐ณ โฒโฒ ) ,๐ ( ๐ง// X ) = ๐ ( ๐ก โฒโฒ // ๐ณ โฒโฒ ) ,๐ ( โ // X ) = ๐ ( โ // ๐ณ โฒโฒ ) โ ๐ ๐ฅ , (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 ๐ณ โฒ and ๐ณ โฒโฒ , andthe probability of frame event ๐ฅ .In the language of e.p.d. of the 1st kind assump-tions (6.6) mean that ๐ ( ๐ โฒ // ๐ณ โฒ ) + ๐ ( ๐ โฒ ๐ก โฒ // ๐ณ โฒ )++ ๐ ( ๐ โฒโฒ // ๐ณ โฒ ) + ๐ ( ๐ โฒโฒ ๐ก โฒโฒ // ๐ณ โฒ ) = ๐ ๐ฆ ,๐ ( ๐ก โฒ // ๐ณ โฒ ) + ๐ ( ๐ โฒ ๐ก โฒ // ๐ณ โฒ )++ ๐ ( ๐ก โฒโฒ // ๐ณ โฒ ) + ๐ ( ๐ โฒโฒ ๐ก โฒโฒ // ๐ณ โฒ ) = ๐ ๐ง , (6.12)or the same in the language of probabilities events: ๐ ๐ โฒ + ๐ ๐ โฒโฒ = ๐ ๐ฆ ,๐ ๐ก โฒ + ๐ ๐ก โฒโฒ = ๐ ๐ง , (6.13)In addition, the third pair of inclusions under theassumptions (6.6) means that ๐ ๐ โฒ + ๐ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ (cid:54) ๐ ๐ฅ ,๐ ๐ โฒโฒ + ๐ ๐ก โฒโฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ (cid:54) โ ๐ ๐ฅ , (6.14)where ๐ ๐ โฒ ๐ก โฒ = ๐ ( ๐ โฒ ๐ก โฒ // ๐ณ โฒ ) = P ( ๐ โฒ โฉ ๐ก โฒ ) ,๐ ๐ โฒโฒ ๐ก โฒโฒ = ๐ ( ๐ โฒโฒ ๐ก โฒโฒ // ๐ณ โฒโฒ ) = P ( ๐ โฒโฒ โฉ ๐ก โฒโฒ ) (6.15)are probabilities of double intersections of eventsfrom the inserted doublets ๐ณ โฒ and ๐ณ โฒโฒ .Taking into account the Frยดechet inequalities the re-strictions (6.13) and (6.14) are equivalent to thefollowing inequalities for 4 parameters ๐ ๐ โฒ , ๐ ๐ก โฒ , ๐ ๐ โฒ ๐ก โฒ ๐ ๐ โฒโฒ ๐ก โฒโฒ of inserted doublets ๐ณ โฒ and ๐ณ โฒโฒ : (cid:54) ๐ ๐ โฒ (cid:54) ๐ ๐ฆ , (cid:54) ๐ ๐ก โฒ (cid:54) ๐ ๐ง ,๐ โ ๐ โฒ ๐ก โฒ (cid:54) ๐ ๐ โฒ ๐ก โฒ (cid:54) ๐ + ๐ โฒ ๐ก โฒ ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ (cid:54) ๐ ๐ โฒโฒ ๐ก โฒโฒ (cid:54) ๐ + ๐ โฒโฒ ๐ก โฒโฒ , (6.16)where ๐ โ ๐ โฒ ๐ก โฒ = max { , ๐ ๐ โฒ + ๐ ๐ก โฒ โ ๐ ๐ฅ } ,๐ + ๐ โฒ ๐ก โฒ = min { ๐ ๐ โฒ , ๐ ๐ก โฒ } ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ = max { , ๐ ๐ฅ + ๐ ๐ฆ + ๐ ๐ง โ โ ๐ ๐ โฒ โ ๐ ๐ก โฒ } ,๐ + ๐ โฒโฒ ๐ก โฒโฒ = min { ๐ ๐ฆ โ ๐ ๐ โฒ , ๐ ๐ง โ ๐ ๐ก โฒ } , (6.17)are the lower and upper Frechet-boundaries ofprobabilities of double intersections of inserteddoublets ๐ณ โฒ and ๐ณ โฒโฒ with respect to the frame mono-plet { ๐ฅ } .Letโs write the formulas (6.11), using only these4 parameters and remembering the restrictions OROBYEV (6.16): ๐ ( ๐ฅ๐ฆ๐ง// X ) = ๐ ๐ โฒ ๐ก โฒ ,๐ ( ๐ฅ๐ฆ// X ) = ๐ ๐ โฒ โ ๐ ๐ โฒ ๐ก โฒ ,๐ ( ๐ฅ๐ง// X ) = ๐ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ ,๐ ( ๐ฅ// X ) = ๐ ๐ฅ โ ๐ ๐ โฒ โ ๐ ๐ก โฒ + ๐ ๐ โฒ ๐ก โฒ ,๐ ( ๐ฆ๐ง// X ) = ๐ ๐ โฒโฒ ๐ก โฒโฒ ,๐ ( ๐ฆ// X ) = ๐ ๐ฆ โ ๐ ๐ โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ,๐ ( ๐ง// X ) = ๐ ๐ง โ ๐ ๐ก โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ,๐ ( โ // X ) = 1 โ ๐ ๐ฅ โ ๐ ๐ฆ โ ๐ ๐ง + ๐ ๐ โฒ + ๐ ๐ก โฒ + ๐ ๐ โฒโฒ ๐ก โฒโฒ . (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 = { ๐ฅ, ๐ฆ, ๐ง } = { ๐ฅ } + { ๐ฆ, ๐ง } = { ๐ฅ } + ๐ณ : K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + { ๐ฅ } ) )๏ธ == โงโชโจโชโฉ K ( { ๐ฅ } ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒ ) )๏ธ , ๐ = { ๐ฅ } , K ( โ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) )๏ธ , ๐ = โ , (6.19) K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + { ๐ฅ } ) )๏ธ == โงโชโจโชโฉ K |{ ๐ฅ } (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |{ ๐ฅ } // { ๐ฅ } ) )๏ธ ๐ ๐ฅ , ๐ = { ๐ฅ } , K |โ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |โ // { ๐ฅ } ) )๏ธ (1 โ ๐ ๐ฅ ) , ๐ = โ . (6.20)In the formulas (6.19) the inserted pseudo-Kopulas K ( { ๐ฅ } ) and K ( โ ) are de๏ฌned by the ๏ฌrst and the sec-ond four probabilities from (6.18) correspondingly,i.e., by the formulas: K ( { ๐ฅ } ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒ ) )๏ธ == โงโชโชโชโจโชโชโชโฉ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ฆ, ๐ง } ,๐ ๐ โฒ โ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ฆ } ,๐ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ง } ,๐ ๐ฅ โ ๐ ๐ โฒ โ ๐ ๐ก โฒ + ๐ ๐ โฒ ๐ก โฒ , ๐ = โ , (6.21) K ( โ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) )๏ธ == โงโชโชโชโชโชโชโจโชโชโชโชโชโชโฉ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ฆ, ๐ง } ,๐ ๐ฆ โ ๐ ๐ โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ฆ } ,๐ ๐ง โ ๐ ๐ก โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ง } , โ ๐ ๐ฅ โ ๐ ๐ฆ โ ๐ ๐ง ++ ๐ ๐ โฒ + ๐ ๐ก โฒ + ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = โ , (6.22)where, for example, for ๐ = { ๐ฆ, ๐ง }{ ๐ฆ, ๐ง } ( โฉ{ ๐ฅ } // { ๐ฅ } ) = { ๐ฆ โฉ ๐ฅ, ๐ง โฉ ๐ฅ } โ ๐ณ โฒ = { ๐ โฒ , ๐ก โฒ } , { ๐ฆ, ๐ง } ( โฉโ // { ๐ฅ } ) = { ๐ฆ โฉ ๐ฅ ๐ , ๐ง โฉ ๐ฅ ๐ } โ ๐ณ โฒโฒ = { ๐ โฒโฒ , ๐ก โฒโฒ } , (6.23) and the corresponding sets of marginal probabili-ties of inserted doublets ๐ณ โฒ ๐ณ โฒโฒ have the form ห ๐ ( ๐ |{ ๐ฆ,๐ง } ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒโฒ ) = { ๐ ๐ โฒ , ๐ ๐ก โฒ } , ห ๐ ( ๐ |{ ๐ฆ,๐ง } ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) = { ๐ ๐ฆ โ ๐ ๐ โฒ , ๐ ๐ง โ ๐ ๐ก โฒ } . (6.24)In the formulas (6.20) the conditional Kopulas K |{ ๐ฅ } and K |โ are de๏ฌned by the ๏ฌrst and the second fourprobabilities from (6.18), normalized by ๐ ๐ฅ and by โ ๐ ๐ฅ correspondingly, i.e., by the formulas: K |{ ๐ฅ } (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |{ ๐ฅ } // { ๐ฅ } ) )๏ธ == โงโชโชโชโชโจโชโชโชโชโฉ ๐ ๐ฅ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ฆ, ๐ง } , ๐ ๐ฅ ( ๐ ๐ โฒ โ ๐ ๐ โฒ ๐ก โฒ ) , ๐ = { ๐ฆ } , ๐ ๐ฅ ( ๐ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ ) , ๐ = { ๐ง } , โ ๐ ๐ฅ ( ๐ ๐ โฒ + ๐ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ ) , ๐ = โ , (6.25) K |โ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |โ // { ๐ฅ } ) )๏ธ == โงโชโชโชโชโชโชโจโชโชโชโชโชโชโฉ โ ๐ ๐ฅ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ฆ, ๐ง } , โ ๐ ๐ฅ ( ๐ ๐ฆ โ ๐ ๐ โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ) , ๐ = { ๐ฆ } , โ ๐ ๐ฅ ( ๐ ๐ง โ ๐ ๐ก โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ) , ๐ = { ๐ง } , โ ๐ ๐ฅ (1 โ ๐ ๐ฅ โ ๐ ๐ฆ โ ๐ ๐ง )++ โ ๐ ๐ฅ ( ๐ ๐ โฒ + ๐ ๐ก โฒ + ๐ ๐ โฒโฒ ๐ก โฒโฒ ) , ๐ = โ . (6.26)The corresponding sets of marginal conditionalprobabilities of events ๐ฆ, ๐ง โ ๐ณ with respectto the frame terraced events ter ( { ๐ฅ } // { ๐ฅ } ) = ๐ฅ ter ( โ // { ๐ฅ } ) = ๐ฅ ๐ correspondingly have the from: ห ๐ ( ๐ |๐ณ // ๐ณ |{ ๐ฅ } // { ๐ฅ } ) = {๏ธ ๐ ๐ โฒ ๐ ๐ฅ , ๐ ๐ก โฒ ๐ ๐ฅ }๏ธ , ห ๐ ( ๐ |๐ณ // ๐ณ |โ // { ๐ฅ } ) = {๏ธ ๐ ๐ฆ โ ๐ ๐ โฒ โ ๐ ๐ฅ , ๐ ๐ง โ ๐ ๐ก โฒ โ ๐ ๐ฅ }๏ธ . (6.27)Remind, that the four functional parameters ๐ ๐ โฒ , ๐ ๐ก โฒ , ๐ ๐ โฒ ๐ก โฒ and ๐ ๐ โฒโฒ ๐ก โฒโฒ 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 ห ๐ - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered frame half-rare quadruplet of events X = { ๐ฅ, ๐ฆ, ๐ง, ๐ฃ } with the X -set of marginal probabilities ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง , ๐ ๐ฃ } , where / (cid:62) ๐ ๐ฅ (cid:62) ๐ ๐ฆ (cid:62) ๐ ๐ง (cid:62) ๐ ๐ฃ (cid:62) , (7.1)letโs suppose that X = { ๐ฅ } + { ๐ฆ, ๐ง, ๐ฃ } = { ๐ฅ } + ( ๐ณ โฒ (+) ๐ณ โฒโฒ ) (7.2)
02 T HE XIV FAMEMSโ2015 C
ONFERENCE and we have two inserted half-rare triplets of events ๐ณ โฒ = { ๐ โฒ , ๐ก โฒ , ๐ข โฒ } ๐ณ โฒโฒ = { ๐ โฒโฒ , ๐ก โฒโฒ , ๐ข โฒโฒ } , with the known 3-Kopulas, which by de๏ฌnition sat-isfy the following inclusions (see Fig. 11): ๐ โฒ = ๐ฅ โฉ ๐ฆ โ ๐ฅ, ๐ก โฒ = ๐ฅ โฉ ๐ง โ ๐ฅ, ๐ข โฒ = ๐ฅ โฉ ๐ฃ โ ๐ฅ,๐ โฒ โช ๐ก โฒ โช ๐ข โฒ โ ๐ฅ,๐ โฒโฒ = ๐ฅ ๐ โฉ ๐ฆ โ ๐ฅ ๐ , ๐ก โฒโฒ = ๐ฅ ๐ โฉ ๐ง โ ๐ฅ ๐ , ๐ข โฒโฒ = ๐ฅ ๐ โฉ ๐ฃ โ ๐ฅ ๐ ,๐ โฒโฒ โช ๐ก โฒโฒ โช ๐ข โฒโฒ โ ๐ฅ ๐ . (7.3) ฮฉ ter ( ๐ฅ// X ) ter ( โ // X ) ter ( ๐ฅ๐ฆ// X ) ter ( ๐ฆ// X ) ter ( ๐ฅ๐ฆ๐ง// X ) ter ( ๐ฅ๐ฆ๐ฃ// X ) ter ( ๐ฆ๐ง// X ) ter ( ๐ฆ๐ฃ// X ) ter ( ๐ฅ๐ฆ๐ง๐ฃ// X ) ter ( ๐ฆ๐ง๐ฃ// X ) ter ( ๐ฅ๐ง// X ) ter ( ๐ฅ๐ง๐ฃ// X ) ter ( ๐ฅ๐ฃ// X ) ter ( ๐ง// X ) ter ( ๐ง๐ฃ// X ) ter ( ๐ฃ// X ) ๐ฆ โฉ ๐ฅ ๐ฆ โฉ ๐ฅ ๐ ๐ง โฉ ๐ฅ ๐ฃ โฉ ๐ฅ๐ง โฉ ๐ฅ ๐ ๐ฃ โฉ ๐ฅ ๐ โ โ ๐ฅ โ โ ๐ฅ ๐ ฮฉ ๐ โฒ ๐ โฉ ๐ก โฒ ๐ โฉ ๐ข โฒ ๐ โ ๐ฅ๐ ๐ โฒโฒ ๐ โฉ ๐ก โฒโฒ ๐ โฉ ๐ข โฒโฒ ๐ โ ๐ฅ๐ โฒโฉ ๐ก โฒ ๐ โฉ ๐ข โฒ ๐ ๐ โฒโฒโฉ ๐ก โฒโฒ ๐ โฉ ๐ข โฒโฒ ๐๐ โฒโฉ ๐ก โฒโฉ ๐ข โฒ ๐ ๐ โฒโฉ ๐ก โฒ ๐ โฉ ๐ข โฒ ๐ โฒโฒโฉ ๐ก โฒโฒโฉ ๐ข โฒโฒ ๐ ๐ โฒโฒโฉ ๐ก โฒโฒ ๐ โฉ ๐ข โฒโฒ ๐ โฒโฉ ๐ก โฒโฉ ๐ข โฒ ๐ โฒโฒโฉ ๐ก โฒโฒโฉ ๐ข โฒโฒ ๐ โฒ ๐ โฉ ๐ก โฒโฉ ๐ข โฒ ๐ ๐ โฒ ๐ โฉ ๐ก โฒโฉ ๐ข โฒ ๐ โฒ ๐ โฉ ๐ก โฒ ๐ โฉ ๐ข โฒ ๐ โฒโฒ ๐ โฉ ๐ก โฒโฒโฉ ๐ข โฒโฒ ๐ ๐ โฒโฒ ๐ โฉ ๐ก โฒโฒโฉ ๐ข โฒโฒ ๐ โฒโฒ ๐ โฉ ๐ก โฒโฒ ๐ โฉ ๐ข โฒโฒ ๐ โฒ ๐ โฒโฒ ๐ก โฒ ๐ข โฒ ๐ก โฒโฒ ๐ข โฒโฒ โ โ ๐ฅ โ โ ๐ฅ ๐ Figure 11: Venn diagrams of the frame half-rare quadruplet of events X = { ๐ฅ, ๐ฆ, ๐ง, ๐ฃ } , / (cid:62) ๐ ๐ฅ (cid:62) ๐ ๐ฆ (cid:62) ๐ ๐ง (cid:62) ๐ ๐ฃ (up), and two insertedtriplets ๐ณ โฒ = { ๐ โฒ , ๐ก โฒ , ๐ข โฒ } and ๐ณ โฒโฒ = { ๐ โฒโฒ , ๐ก โฒโฒ , ๐ข โฒโฒ } (down), agreed with theframe quadruplet X in the following sense: ๐ฆ = ๐ โฒ + ๐ โฒโฒ , ๐ง = ๐ก โฒ + ๐ก โฒโฒ , ๐ฃ = ๐ข โฒ + ๐ข โฒโฒ ๐ โฒ โช ๐ก โฒ โช ๐ข โฒ โ ๐ฅ, ๐ โฒโฒ โช ๐ก โฒโฒ โช ๐ข โฒโฒ โ ๐ฅ ๐ . The recurrent formulas, which express the e.p.d.of the 1st kind of ห ๐ -ordered half-rare quadruplet ofevents X through the e.p.d. of the 1st kind of twoinserted triplets ๐ณ โฒ and ๐ณ โฒโฒ , 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 Frยดechet-restrictionsfor
11 = 2 โ โ functional parameters of a Kop-ula of quadruplet of events, to derive the recur-rent sequence of such Frยดechet-restrictions, which begins with Frยดechet-restrictions for a doublet ofevents (5.36), continues with Frยดechet-restrictionsfor a triplet of events (6.16), and should be sup-ported by Frยดechet-restrictions for parameters of aKopula of quadruplet of events X = { ๐ฅ, ๐ฆ, ๐ง, ๐ฃ } andso on.To this end, we ๏ฌrst recall Frยดechet-restrictions forparameters of Kopulas of a doublet and a triplet ofevents. For a Kopula of doublet of events, the Frยดechet-restrictions of a โ โ parameter of insertedmonoplets ๐ณ โฒ and ๐ณ โฒโฒ have the form: (cid:54) ๐ ๐ โฒ (cid:54) ๐ ๐ฆ . (7.4) For a Kopula of doublet of events, the Frยดechet-restrictions of โ โ parameters of inserteddoublets ๐ณ โฒ and ๐ณ โฒโฒ have the form: (cid:54) ๐ ๐ โฒ (cid:54) ๐ ๐ฆ , (cid:54) ๐ ๐ก โฒ (cid:54) ๐ ๐ง ,๐ โ ๐ โฒ ๐ก โฒ (cid:54) ๐ ๐ โฒ ๐ก โฒ (cid:54) ๐ + ๐ โฒ ๐ก โฒ ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ (cid:54) ๐ ๐ โฒโฒ ๐ก โฒโฒ (cid:54) ๐ + ๐ โฒโฒ ๐ก โฒโฒ , (7.5)where ๐ โ ๐ โฒ ๐ก โฒ = max { , ๐ ๐ โฒ + ๐ ๐ก โฒ โ ๐ ๐ฅ } ,๐ + ๐ โฒ ๐ก โฒ = min { ๐ ๐ โฒ , ๐ ๐ก โฒ } ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ = max { , ๐ ๐ฅ + ๐ ๐ฆ + ๐ ๐ง โ โ ๐ ๐ โฒ โ ๐ ๐ก โฒ } ,๐ + ๐ โฒโฒ ๐ก โฒโฒ = min { ๐ ๐ฆ โ ๐ ๐ โฒ , ๐ ๐ง โ ๐ ๐ก โฒ } , (7.6)are the lower and upper Frยดechet-boundaries prob-abilities of double intersections of events from in-serted doublets ๐ณ โฒ and ๐ณ โฒโฒ with respect to the framemonoplet { ๐ฅ } .The case of triplet of events gives a newlevel of Frยดechet-restrictions (the two last Frยดechet-boundaries in (7.6)), when probabilities of dou-ble intersections of events from inserted doubletshave Frยดechet-boundaries that depend not only onmarginal probabilities of the triplet, but and on in-serted marginal probabilities on which, in turn, theusual Frยดechet-restrictions mentioned above are im-posed. For a Kopula of quadruplet of events the Frยดechet-restrictions of
11 = 2 โ โ parameters of the in- OROBYEV serted triplet ๐ณ โฒ and ๐ณ โฒโฒ have the form: (cid:54) ๐ ๐ โฒ (cid:54) ๐ ๐ฆ , (cid:54) ๐ ๐ก โฒ (cid:54) ๐ ๐ง , (cid:54) ๐ ๐ข โฒ (cid:54) ๐ ๐ฃ ,๐ โ ๐ โฒ ๐ก โฒ (cid:54) ๐ ๐ โฒ ๐ก โฒ (cid:54) ๐ + ๐ โฒ ๐ก โฒ ,๐ โ ๐ โฒ ๐ข โฒ (cid:54) ๐ ๐ โฒ ๐ข โฒ (cid:54) ๐ + ๐ โฒ ๐ข โฒ ,๐ โ ๐ก โฒ ๐ข โฒ (cid:54) ๐ ๐ก โฒ ๐ข โฒ (cid:54) ๐ + ๐ก โฒ ๐ข โฒ ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ (cid:54) ๐ ๐ โฒโฒ ๐ก โฒโฒ (cid:54) ๐ + ๐ โฒโฒ ๐ก โฒโฒ ,๐ โ ๐ โฒโฒ ๐ข โฒโฒ (cid:54) ๐ ๐ โฒโฒ ๐ข โฒโฒ (cid:54) ๐ + ๐ โฒโฒ ๐ข โฒโฒ ,๐ โ ๐ก โฒโฒ ๐ข โฒโฒ (cid:54) ๐ ๐ก โฒโฒ ๐ข โฒโฒ (cid:54) ๐ + ๐ก โฒโฒ ๐ข โฒโฒ ,๐ โ ๐ โฒ ๐ก โฒ ๐ข โฒ (cid:54) ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ (cid:54) ๐ + ๐ โฒ ๐ก โฒ ๐ข โฒ ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ (cid:54) ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ (cid:54) ๐ + ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ , (7.7)where ๐ โ ๐ โฒ ๐ก โฒ = max { , ๐ ๐ โฒ + ๐ ๐ก โฒ โ ๐ ๐ฅ } ,๐ + ๐ โฒ ๐ก โฒ = min { ๐ ๐ โฒ , ๐ ๐ก โฒ } ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ = max { , ๐ ๐ฅ + ๐ ๐ฆ + ๐ ๐ง โ โ ๐ ๐ โฒ โ ๐ ๐ก โฒ } ,๐ + ๐ โฒโฒ ๐ก โฒโฒ = min { ๐ ๐ฆ โ ๐ ๐ โฒ , ๐ ๐ง โ ๐ ๐ก โฒ } ,๐ โ ๐ โฒ ๐ข โฒ = max { , ๐ ๐ โฒ + ๐ ๐ข โฒ โ ๐ ๐ฅ } ,๐ + ๐ โฒ ๐ข โฒ = min { ๐ ๐ โฒ , ๐ ๐ข โฒ } ,๐ โ ๐ โฒโฒ ๐ข โฒโฒ = max { , ๐ ๐ฅ + ๐ ๐ฆ + ๐ ๐ฃ โ โ ๐ ๐ โฒ โ ๐ ๐ข โฒ } ,๐ + ๐ โฒโฒ ๐ข โฒโฒ = min { ๐ ๐ฆ โ ๐ ๐ โฒ , ๐ ๐ฃ โ ๐ ๐ข โฒ } ,๐ โ ๐ก โฒ ๐ข โฒ = max { , ๐ ๐ก โฒ + ๐ ๐ข โฒ โ ๐ ๐ฅ } ,๐ + ๐ก โฒ ๐ข โฒ = min { ๐ ๐ก โฒ , ๐ ๐ข โฒ } ,๐ โ ๐ก โฒโฒ ๐ข โฒโฒ = max { , ๐ ๐ฅ + ๐ ๐ง + ๐ ๐ฃ โ โ ๐ ๐ก โฒ โ ๐ ๐ข โฒ } ,๐ + ๐ก โฒโฒ ๐ข โฒโฒ = min { ๐ ๐ง โ ๐ ๐ก โฒ , ๐ ๐ฃ โ ๐ ๐ข โฒ } ,๐ โ ๐ โฒ ๐ก โฒ ๐ข โฒ = max { , ๐ ๐ โฒ ๐ก โฒ + ๐ ๐ โฒ ๐ข โฒ + ๐ ๐ก โฒ ๐ข โฒ โ ๐ ๐ฅ } ,๐ + ๐ โฒ ๐ก โฒ ๐ข โฒ = min { ๐ ๐ โฒ ๐ก โฒ , ๐ ๐ โฒ ๐ข โฒ , ๐ ๐ก โฒ ๐ข โฒ } ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ = max { , ๐ ๐ โฒโฒ ๐ก โฒโฒ + ๐ ๐ โฒโฒ ๐ข โฒโฒ + ๐ ๐ก โฒโฒ ๐ข โฒโฒ โ โ ๐ ๐ฅ ) } ,๐ + ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ = min { ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ ๐ โฒโฒ ๐ข โฒโฒ , ๐ ๐ก โฒโฒ ๐ข โฒโฒ } (7.8)are the lower and upper Frยดechet-boundaries ofprobabilities of double and triple intersections ofevents from the inserted triplets ๐ณ โฒ and ๐ณ โฒโฒ with re-spect to the frame monoplet { ๐ฅ } .The case of a quadruplet of events gives the fol-lowing level of Frยดechet-restrictions (the four lastFrยดechet-boundaries in (7.8)), when probabilities oftriple intersections of events from inserted tripletshave Frยดechet-boundaries that depend directly notso much on marginal probabilities as on insertedprobabilities of double intersections, on which, inturn, Frโechet-restrictions of the previous level,mentioned above, are imposed.The all Frยดechet-restrictions in the considered framemethods for a doublet, a triplet and a quadru-plet of events differ from the usual Frยดechet-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 Frยดechet-boundaries of their probabilities arefunctions of Frยดechet-boundaries for probabilities ofintersections of lower power.The such Frยดechet-restrictions and Frยดechet-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 Frยดechet-restrictions and recurrentFrยดechet-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 = { ๐ฅ, ๐ฆ, ๐ง, ๐ฃ } = { ๐ฅ } + { ๐ฆ, ๐ง, ๐ฃ } = { ๐ฅ } + ๐ณ : K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + { ๐ฅ } ) )๏ธ == โงโชโจโชโฉ K ( { ๐ฅ } ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒ ) )๏ธ , ๐ = { ๐ฅ } , K ( โ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) )๏ธ , ๐ = โ , (7.9) K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + { ๐ฅ } ) )๏ธ == โงโชโจโชโฉ K |{ ๐ฅ } (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |{ ๐ฅ } // { ๐ฅ } ) )๏ธ ๐ ๐ฅ , ๐ = { ๐ฅ } , K |โ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |โ // { ๐ฅ } ) )๏ธ (1 โ ๐ ๐ฅ ) , ๐ = โ . (7.10)In the formulas (7.9) the inserted pseudo-Kopulas K ( { ๐ฅ } ) and K ( โ ) are de๏ฌned by the octuples of prob-abilities, i.e., by the formulas: K ( { ๐ฅ } ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒ ) )๏ธ == โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ , ๐ = { ๐ฆ, ๐ง, ๐ฃ } ,๐ ๐ โฒ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ , ๐ = { ๐ฆ, ๐ง } ,๐ ๐ โฒ ๐ข โฒ โ ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ , ๐ = { ๐ฆ, ๐ฃ } ,๐ ๐ก โฒ ๐ข โฒ โ ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ , ๐ = { ๐ง, ๐ฃ } ,๐ ๐ โฒ โ ๐ ๐ โฒ ๐ก โฒ โ ๐ ๐ โฒ ๐ข โฒ + ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ , ๐ = { ๐ง } ,๐ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ โ ๐ ๐ก โฒ ๐ข โฒ + ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ , ๐ = { ๐ฃ } ,๐ ๐ข โฒ โ ๐ ๐ โฒ ๐ข โฒ โ ๐ ๐ก โฒ ๐ข โฒ + ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ , ๐ = { ๐ฃ } ,๐ ๐ฅ โ ๐ ๐ โฒ โ ๐ ๐ก โฒ โ ๐ ๐ข โฒ ++ ๐ ๐ โฒ ๐ก โฒ + ๐ ๐ โฒ ๐ข โฒ + ๐ ๐ก โฒ ๐ข โฒ โ ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ , ๐ = โ , (7.11)
04 T HE XIV FAMEMSโ2015 C
ONFERENCE K ( โ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) )๏ธ == โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ , ๐ = { ๐ฆ, ๐ง, ๐ฃ } ,๐ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ , ๐ = { ๐ฆ, ๐ง } ,๐ ๐ โฒโฒ ๐ข โฒโฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ , ๐ = { ๐ฆ, ๐ฃ } ,๐ ๐ก โฒโฒ ๐ข โฒโฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ , ๐ = { ๐ง, ๐ฃ } ,๐ ๐ฆ โ ๐ ๐ โฒ โโ ๐ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ๐ โฒโฒ ๐ข โฒโฒ + ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ , ๐ = { ๐ฆ } ,๐ ๐ง โ ๐ ๐ก โฒ โโ ๐ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ๐ก โฒโฒ ๐ข โฒโฒ + ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ , ๐ = { ๐ง } ,๐ ๐ฃ โ ๐ ๐ก โฒ โโ ๐ ๐ โฒโฒ ๐ข โฒโฒ โ ๐ ๐ก โฒโฒ ๐ข โฒโฒ + ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ , ๐ = { ๐ฃ } , โ ๐ ๐ฅ โ ๐ ๐ฆ โ ๐ ๐ง โ ๐ ๐ฃ ++ ๐ ๐ โฒ + ๐ ๐ก โฒ + ๐ ๐ข โฒ โโ ๐ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ๐ โฒโฒ ๐ข โฒโฒ โ ๐ ๐ก โฒโฒ ๐ข โฒโฒ ++ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ , ๐ = โ , (7.12)where, for example, for ๐ = { ๐ฆ, ๐ง, ๐ฃ }{ ๐ฆ, ๐ง, ๐ฃ } ( โฉ{ ๐ฅ } // { ๐ฅ } ) = { ๐ฆ โฉ ๐ฅ, ๐ง โฉ ๐ฅ, ๐ฃ โฉ ๐ฅ } โ ๐ณ โฒ , { ๐ฆ, ๐ง, ๐ฃ } ( โฉโ // { ๐ฅ } ) = { ๐ฆ โฉ ๐ฅ ๐ , ๐ง โฉ ๐ฅ ๐ , ๐ฃ โฉ ๐ฅ ๐ } โ ๐ณ โฒโฒ , (7.13)and the corresponding sets of marginal probabili-ties of inserted triplets ๐ณ โฒ and ๐ณ โฒโฒ have the form ห ๐ ( ๐ |{ ๐ฆ,๐ง,๐ฃ } ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒโฒ ) = { ๐ ๐ โฒ , ๐ ๐ก โฒ , ๐ ๐ข โฒ } , ห ๐ ( ๐ |{ ๐ฆ,๐ง,๐ฃ } ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) = { ๐ ๐ฆ โ ๐ ๐ โฒ , ๐ ๐ง โ ๐ ๐ก โฒ , ๐ ๐ฃ โ ๐ ๐ข โฒ } . (7.14)In the formulas (7.10) the condirional Kopulas K |{ ๐ฅ } and K |โ are de๏ฌned by the octuples of proba-bilities that are normalized by ๐ ๐ฅ and by โ ๐ ๐ฅ cor-respondingly, i.e., by the formulas: K |{ ๐ฅ } (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |{ ๐ฅ } // { ๐ฅ } ) )๏ธ == โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ ๐ ๐ฅ ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ , ๐ = { ๐ฆ, ๐ง, ๐ฃ } , ๐ ๐ฅ ( ๐ ๐ โฒ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ ) , ๐ = { ๐ฆ, ๐ง } , ๐ ๐ฅ ( ๐ ๐ โฒ ๐ข โฒ โ ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ ) , ๐ = { ๐ฆ, ๐ฃ } , ๐ ๐ฅ ( ๐ ๐ก โฒ ๐ข โฒ โ ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ ) , ๐ = { ๐ง, ๐ฃ } , ๐ ๐ฅ ( ๐ ๐ โฒ โ ๐ ๐ โฒ ๐ก โฒ โ ๐ ๐ โฒ ๐ข โฒ + ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ ) , ๐ = { ๐ง } , ๐ ๐ฅ ( ๐ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ โ ๐ ๐ก โฒ ๐ข โฒ + ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ ) , ๐ = { ๐ฃ } , ๐ ๐ฅ ( ๐ ๐ข โฒ โ ๐ ๐ โฒ ๐ข โฒ โ ๐ ๐ก โฒ ๐ข โฒ + ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ ) ,๐ = { ๐ฃ } , โ ๐ ๐ฅ ( ๐ ๐ โฒ + ๐ ๐ก โฒ + ๐ ๐ข โฒ )++ ๐ ๐ฅ ( ๐ ๐ โฒ ๐ก โฒ + ๐ ๐ โฒ ๐ข โฒ + ๐ ๐ก โฒ ๐ข โฒ ) โโ ๐ ๐ฅ ๐ ๐ โฒ ๐ก โฒ ๐ข โฒ , ๐ = โ , (7.15) K |โ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |โ // { ๐ฅ } ) )๏ธ == โงโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ โ ๐ ๐ฅ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ , ๐ = { ๐ฆ, ๐ง, ๐ฃ } , โ ๐ ๐ฅ ( ๐ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ ) , ๐ = { ๐ฆ, ๐ง } , โ ๐ ๐ฅ ( ๐ ๐ โฒโฒ ๐ข โฒโฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ ) , ๐ = { ๐ฆ, ๐ฃ } , โ ๐ ๐ฅ ( ๐ ๐ก โฒโฒ ๐ข โฒโฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ ) , ๐ = { ๐ง, ๐ฃ } , โ ๐ ๐ฅ ( ๐ ๐ฆ โ ๐ ๐ โฒ ) โโ โ ๐ ๐ฅ ( ๐ ๐ โฒโฒ ๐ก โฒโฒ + ๐ ๐ โฒโฒ ๐ข โฒโฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ ) , ๐ = { ๐ฆ } , โ ๐ ๐ฅ ( ๐ ๐ง โ ๐ ๐ก โฒ ) โโ โ ๐ ๐ฅ ( ๐ ๐ โฒโฒ ๐ก โฒโฒ + ๐ ๐ก โฒโฒ ๐ข โฒโฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ ) , ๐ = { ๐ง } , โ ๐ ๐ฅ ( ๐ ๐ฃ โ ๐ ๐ก โฒ ) โโ โ ๐ ๐ฅ ( ๐ ๐ โฒโฒ ๐ข โฒโฒ + ๐ ๐ก โฒโฒ ๐ข โฒโฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ ) , ๐ = { ๐ฃ } , โ โ ๐ ๐ฅ ( ๐ ๐ฆ + ๐ ๐ง + ๐ ๐ฃ )++ โ ๐ ๐ฅ ( ๐ ๐ โฒ + ๐ ๐ก โฒ + ๐ ๐ข โฒ ) โโ โ ๐ ๐ฅ ( ๐ ๐ โฒโฒ ๐ก โฒโฒ + ๐ ๐ โฒโฒ ๐ข โฒโฒ + ๐ ๐ก โฒโฒ ๐ข โฒโฒ )++ โ ๐ ๐ฅ ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ , ๐ = โ , (7.16)The corresponding sets of marginal conditionalprobabilities of events ๐ฆ, ๐ง โ ๐ณ with respect tothe frame terraced events ter ( { ๐ฅ } // { ๐ฅ } ) = ๐ฅ and ter ( โ // { ๐ฅ } ) = ๐ฅ ๐ correspondingly have the form: ห ๐ ( ๐ |๐ณ // ๐ณ |{ ๐ฅ } // { ๐ฅ } ) = {๏ธ ๐ ๐ โฒ ๐ ๐ฅ , ๐ ๐ก โฒ ๐ ๐ฅ , ๐ ๐ข โฒ ๐ ๐ฅ }๏ธ , ห ๐ ( ๐ |๐ณ // ๐ณ |โ // { ๐ฅ } ) = {๏ธ ๐ ๐ฆ โ ๐ ๐ โฒ โ ๐ ๐ฅ , ๐ ๐ง โ ๐ ๐ก โฒ โ ๐ ๐ฅ , ๐ ๐ฃ โ ๐ ๐ข โฒ โ ๐ ๐ฅ }๏ธ . (7.17)Recall that 11 functional parameters ๐ ๐ โฒ , ๐ ๐ก โฒ , ๐ ๐ข โฒ ,๐ ๐ โฒ ๐ก โฒ , ๐ ๐ โฒ ๐ข โฒ , ๐ ๐ก โฒ ๐ข โฒ ,๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ ๐ โฒโฒ ๐ข โฒโฒ , ๐ ๐ก โฒโฒ ๐ข โฒโฒ ,๐ ๐ โฒ ๐ก โฒ ๐ข โฒ , ๐ ๐ โฒโฒ ๐ก โฒโฒ ๐ข โฒโฒ (7.18)in the recurrent formulas (7.11, 7.12) and (7.15,7.16) obey the Frยดechet-restrictions (7.7) andFrยดechet-boundaries (7.8). ๐ -Kopula First, without the frame method, which is not re-quired here, letโs consider the simplest exampleof the ๐ -Kopula K โ ฮจ X of an ๐ -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 , โ X โ [0 , , that for all ๐ง โ X โ๏ธ ๐ฅ โ ๐ โ X K (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = ๐ค ๐ฅ . A such simplest example of a 1-function on the unit X -hypercube is the so-called independent ๐ -Kopula OROBYEV which for all free variables ห ๐ค โ [0 , โ X is de๏ฌnedby the formula: K ( ห ๐ค ) = โ๏ธ ๐ฅ โ X ๐ค ๐ฅ , (8.1)that provides it on each ( ๐ | ห ๐ค ) -phenomenon-domthe following ๐ values: K (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = โ๏ธ ๐ฅ โ ๐ ๐ค ๐ฅ โ๏ธ ๐ฅ โ X โ ๐ (1 โ ๐ค ๐ฅ ) (8.2)for ๐ โ X . Indeed as in the case of doublet ofevents this function is a 1-function, since for all ๐ฅ โ X โ๏ธ ๐ฅ โ ๐ โ X (๏ธ โ๏ธ ๐ง โ ๐ ๐ค ๐ง โ๏ธ ๐ง โ X โ ๐ (1 โ ๐ค ๐ง ) )๏ธ = ๐ค ๐ฅ . The e.p.d. of the 1st kind of ondependent ๐ -s.e. X with the X -set of probabilities of events ห ๐ is de๏ฌnedby ๐ values of the independent ๐ -Kopula (8.1) onthe ( ๐ | ห ๐ ) -phenomenon-dom by the general formu-las of half-rare variables, i.e., for ๐ โ { ๐ฅ, ๐ฆ } : ๐ ( ๐// X ) = K (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ = โ๏ธ ๐ฅ โ ๐ ๐ ๐ฅ โ๏ธ ๐ฅ โ X โ ๐ (1 โ ๐ ๐ฅ ) . (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 (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + { ๐ฅ } ) )๏ธ == โงโชโจโชโฉ K ( { ๐ฅ } ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒ ) )๏ธ , ๐ = { ๐ฅ } , K ( โ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) )๏ธ , ๐ = โ , (8.4) K (๏ธ ห ๐ ( ๐ | ๐ + ๐// ๐ณ + { ๐ฅ } ) )๏ธ == โงโชโจโชโฉ K |{ ๐ฅ } (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |{ ๐ฅ } // { ๐ฅ } ) )๏ธ ๐ , ๐ = { ๐ฅ } , K |โ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |โ // { ๐ฅ } ) )๏ธ (1 โ ๐ ) , ๐ = โ , (8.5)which express the ๐ -Kopula of ๐ -s.e. X = ๐ณ + { ๐ฅ } ,where ๐ณ = ๐ณ โฒ (+) ๐ณ โฒโฒ , through the known probabil-ity ๐ of the event ๐ฅ and together with it eithertwo known inserted pseudo- ( ๐ โ -Kopulas (see Def-inition 9), i.e., pseudo- ( ๐ โ -Kopulas of inserted ( ๐ โ -s.e.โs ๐ณ โฒ and ๐ณ โฒโฒ in the frame monoplet { ๐ฅ } ,or through two known conditional ( ๐ โ -Kopulas (see De๏ฌnition 10) with respect to the frame mono-plet { ๐ฅ } for the same ๐ณ โฒ and ๐ณ โฒโฒ . Perhaps this statement deserves to be called a lemma,which, incidentally, is not di๏ฌcult to prove.
We write out more detailed formulas for corre-sponding pseudo- ( ๐ โ -Kopulas: K ( { ๐ฅ } ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉ{ ๐ฅ } // { ๐ฅ } ) // ๐ณ โฒ ) )๏ธ = K ( { ๐ฅ } ) (๏ธ ห ๐ ( ๐ | ๐ โฒ // ๐ณ โฒ ) )๏ธ == {๏ธ ๐ ( ๐ โฒ // ๐ณ โฒ ) , ๐ โฒ ฬธ = โ ,๐ ( โ // ๐ณ โฒ ) โ ๐ , ๐ โฒ = โ , K ( โ ) (๏ธ ห ๐ ( ๐ | ๐ ( โฉโ // { ๐ฅ } ) // ๐ณ โฒโฒ ) )๏ธ = K ( โ ) (๏ธ ห ๐ ( ๐ | ๐ โฒโฒ // ๐ณ โฒโฒ ) )๏ธ == {๏ธ ๐ ( ๐ โฒโฒ // ๐ณ โฒโฒ ) , ๐ โฒโฒ ฬธ = โ ,๐ ( โ // ๐ณ โฒโฒ ) โ ๐ , ๐ โฒโฒ = โ , (8.6)and for conditional ( ๐ โ -Kopulas: K |{ ๐ฅ } (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |{ ๐ฅ } // { ๐ฅ } ) )๏ธ == {๏ธ ๐ ๐ ( ๐ โฒ // ๐ณ โฒ ) , ๐ โฒ ฬธ = โ , ๐ ( ๐ ( โ // ๐ณ โฒ ) โ ๐ ) , ๐ โฒ = โ , K |โ (๏ธ ห ๐ ( ๐ | ๐//
๐ณ |โ // { ๐ฅ } ) )๏ธ == {๏ธ โ ๐ ๐ ( ๐ โฒโฒ // ๐ณ โฒโฒ ) , ๐ โฒโฒ ฬธ = โ , โ ๐ ( ๐ ( โ // ๐ณ โฒโฒ ) โ ๐ ) , ๐ โฒโฒ = โ , (8.7)where ๐ โฒ = ๐ ( โฉ{ ๐ฅ } // { ๐ฅ } ) = ๐ ( โฉ ) { ๐ฅ } = { ๐ฅ โฉ ๐ฅ , ๐ฅ โ ๐ } ๐ โฒโฒ = ๐ ( โฉโ // { ๐ฅ } ) = ๐ ( โฉ ) { ๐ฅ ๐ } = { ๐ฅ โฉ ๐ฅ ๐ , ๐ฅ โ ๐ } ๐ โ ๐ณ . Now let us consider recurrent formulas forFrยดechet-boundaries Frยดechet-restrictions and forthe ๐ โ ๐ โ functional parameters of an ๐ -Kopulaof ๐ -set of events X = { ๐ฅ , ๐ฅ , . . . , ๐ฅ ๐ โ } == { ๐ฅ } + { ๐ฅ , . . . , ๐ฅ ๐ โ } == { ๐ฅ } + ๐ณ == { ๐ฅ } + ( ๐ณ โฒ (+) ๐ณ โฒโฒ ) , (8.8)where ๐ณ โฒ = { ๐ฅ โฉ ๐ฅ , . . . , ๐ฅ โฉ ๐ฅ ๐ โ } , ๐ณ โฒโฒ = { ๐ฅ ๐ โฉ ๐ฅ , . . . , ๐ฅ ๐ โฉ ๐ฅ ๐ โ } (8.9)are inserted ( ๐ โ -s.e.โs, and ห ๐ ( ๐ | X // X ) = { ๐ , ๐ , . . . , ๐ ๐ โ } (8.10)is the X -set of probabilities of marginal events from X , i.e., ๐ ๐ = P ( ๐ฅ ๐ ) , ๐ = 0 , , . . . , ๐ โ .Judging by the form of Frยดechet-boundariesFrยดechet-restrictions for a doublet, a triplet and
06 T HE XIV FAMEMSโ2015 C
ONFERENCE a quadruplet of events, collected in paragraph 7.2,these Frยดechet-restrictions consists of two groups,such that one of them, which refers to the parame-ters of the inserted ( ๐ โ -s.e. ๐ณ โฒ , consists of ๐ โ โ Frยดechet-restrictions, and the other, which refersto the parameters of the inserted ( ๐ โ -s.e. ๐ณ โฒโฒ ,consists of ๐ โ โ ( ๐ โ โ Frยดechet-restrictions.And, as it should: ๐ โ ๐ โ ๐ โ โ ๐ โ โ ( ๐ โ โ . (8.11)The ๏ฌrst group, related to the inserted ( ๐ โ -s.e. ๐ณ โฒ , contains Frยดechet-restrictions for probabilitiesof the second kind ๐ ๐ โฒ // ๐ณ โฒ = P (๏ธ โ๏ธ ๐ฅ โฒ โ ๐ โฒ ๐ฅ โฒ )๏ธ , (8.12)that are numbered by nonempty subsets ๐ โฒ ฬธ = โ ofinserted ( ๐ โ -s.e. ๐ณ โฒ (the number od such subsets: ๐ โ โ ); the second group, related to the inserted ( ๐ โ -s.e. ๐ณ โฒโฒ , contains Frยดechet-restrictions for thesuch probabilities of the second kind: ๐ ๐ โฒโฒ // ๐ณ โฒโฒ = P (๏ธ โ๏ธ ๐ฅ โฒโฒ โ ๐ โฒโฒ ๐ฅ โฒโฒ )๏ธ , (8.13)that are numbered by subsets ๐ โฒโฒ โ ๐ณ โฒโฒ with thepower | ๐ โฒโฒ | (cid:62) (number of such subsets: ๐ โ โ ( ๐ โ โ ). Note 18 (denotations for subsets of fixed power).
To more conveniently represent the recurrentFrยดechet-restrictions, agree to denote ๐ โฒ ๐ โ ๐ณ โฒ โโ | ๐ โฒ ๐ | = ๐,๐ โฒโฒ ๐ โ ๐ณ โฒโฒ โโ | ๐ โฒโฒ ๐ | = ๐. (8.14)the subsets consisting of ๐ events. In this nota-tion, for example, the X โset of marginal probabil-ities ห ๐ ( ๐ | X // X ) is written as the X -set probabilities ofthe second kind that are numbered by monopletsof events ๐ = { ๐ฅ } , ๐ฅ โ X : { ๐ , ๐ , . . . , ๐ ๐ โ } = { ๐ ๐ // X , ๐ โ X } . (8.15)The set of probabilities of double intersection ofevents ๐ฅ โ X , i.e., the set of probabilities of the sec-ond kind that are numbered by doublets, has theform: { ๐ { ๐ฅ,๐ฆ } // X , { ๐ฅ, ๐ฆ } โ X } = { ๐ ๐ // X , ๐ โ X } . (8.16)And the set of probabilities of triple intersectionsof events ๐ฅ โ X , i.e., the set of probabilities of thesecond kind that are numbered by triplets, has theform: { ๐ { ๐ฅ,๐ฆ,๐ง } // X , { ๐ฅ, ๐ฆ, ๐ง } โ X } = { ๐ ๐ // X , ๐ โ X } (8.17)and so on. Note 19 (recurrent formulas for Frยดechet-boundaries and Frยดechet-restrictions).
Probabilitiesof ๐ -intersections ( ๐ = 2 , ..., ๐ โ of events from theinserted s.e.โs ๐ณ โฒ and ๐ณ โฒโฒ have the recurrent Frยดechet-restrictions (see paragraph 7.2) that are writtenby denotations from Note 18 by the following waywith respect to ๐ณ โฒ : ๐ โ ๐ โฒ ๐ // ๐ณ โฒ (cid:54) ๐ ๐ โฒ ๐ // ๐ณ โฒ (cid:54) ๐ + ๐ โฒ ๐ // ๐ณ โฒ , (8.18)where ๐ โ ๐ โฒ ๐ // ๐ณ โฒ = max โงโจโฉ , ๐ ๐ฅ โ โ๏ธ ๐ โฒ ๐ โ โ ๐ โฒ ๐ ( ๐ ๐ฅ โ ๐ ๐ โฒ ๐ โ // ๐ณ โฒ ) โซโฌโญ ,๐ + ๐ โฒ ๐ // ๐ณ โฒ = min ๐ โฒ ๐ โ โ ๐ โฒ ๐ {๏ธ ๐ ๐ โฒ ๐ โ // ๐ณ โฒ , ๐ โฒ ๐ โ โ ๐ณ โฒ }๏ธ . (8.19)are recurrent the lower and upper Frยดechet-boundaries . And the lower Frยดechet-boundarywe can write somewhat differently after simpletransformations: ๐ โ ๐ โฒ ๐ // ๐ณ โฒ = max โงโจโฉ , โ๏ธ ๐ โฒ ๐ โ โ ๐ โฒ ๐ ๐ ๐ โฒ ๐ โ // ๐ณ โฒ โ ( ๐ โ ๐ ๐ฅ โซโฌโญ . (8.20)Similar look the recurrent Frยดechet-restrictions withrespect to the inserted s.e. ๐ณ โฒโฒ : ๐ โ ๐ โฒโฒ ๐ // ๐ณ โฒโฒ (cid:54) ๐ ๐ โฒโฒ ๐ // ๐ณ โฒโฒ (cid:54) ๐ + ๐ โฒโฒ ๐ // ๐ณ โฒโฒ , (8.21)where ๐ โ ๐ โฒโฒ ๐ // ๐ณ โฒโฒ = max โงโจโฉ , โ ๐ ๐ฅ โ โ๏ธ ๐ โฒโฒ ๐ โ โ ๐ โฒโฒ ๐ (1 โ ๐ ๐ฅ โ ๐ ๐ โฒโฒ ๐ โ // ๐ณ โฒโฒ ) โซโฌโญ ,๐ + ๐ โฒโฒ ๐ // ๐ณ โฒโฒ = min ๐ โฒโฒ ๐ โ โ ๐ โฒโฒ ๐ {๏ธ ๐ ๐ โฒโฒ ๐ โ // ๐ณ โฒโฒ , ๐ โฒโฒ ๐ โ โ ๐ณ โฒโฒ }๏ธ . (8.22)are recurrent the lower and upper Frยดechet-boundaries . And the lower Frยดechet-boundarywe can write somewhat differently after simpletransformations: ๐ โ ๐ โฒโฒ ๐ // ๐ณ โฒโฒ == max โงโจโฉ , โ๏ธ ๐ โฒโฒ ๐ โ โ ๐ โฒโฒ ๐ ๐ ๐ โฒโฒ ๐ โ // ๐ณ โฒโฒ โ ( ๐ โ โ ๐ ๐ฅ ) โซโฌโญ . (8.23)It remains to write out more ๐ โ recurrent Frยดechet-restrictions on probabilities of marginal eventsfrom the inserted s.e. ๐ณ โฒ , i.e., on probabilities ofthe second kind that are numbered by monoplets ๐ โฒ โ ๐ณ โฒ : (cid:54) ๐ ๐ โฒ // ๐ณ โฒ (cid:54) ๐ ๐ // ๐ณ , (8.24)which are restricted by marginal probabilities ofevents from the ( ๐ โ -s.e. ๐ณ and which to-gether with the recurrent Frยดechet-restrictions (8.18, OROBYEV recurrent Frยดechet-restrictions . This totality consists of ๐ โ ๐ โ restric-tions. And recurrent the lower and upper Frยดechet-boundaries in these restrictions are de๏ฌned by re-current formulas (8.19, 8.22). Letโs consider parametrization on an example offunctional parameters ๐ ๐ โฒ , ๐ ๐ก โฒ , ๐ ๐ โฒ ๐ก โฒ ๐ ๐ โฒโฒ ๐ก โฒโฒ of 3-Kopulaof the ห ๐ -ordered half-rare triplet X = { ๐ฅ, ๐ฆ, ๐ง } , whichin the frame method is constructed from two in-serted pseudo-2-Kopulas. ๐ ๐ โฒ and ๐ ๐ก โฒ The Frยดechet-restriction of the functional parameter ๐ ๐ โฒ = ๐ ๐ โฒ ( ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง ) 0 (cid:54) ๐ ๐ โฒ (cid:54) ๐ ๐ฆ , (9.1)that in the frame method has a sense of probabilityof double intersection of events ๐ฅ and ๐ฆ : ๐ ๐ โฒ = ๐ ๐ฅ๐ฆ// X = P ( ๐ฅ โฉ ๐ฆ ) , (9.2)is baced on the notion of Frยดechet-correlation [1] Kor ๐ฅ๐ฆ = โงโชโจโชโฉ Kov ๐ฅ๐ฆ | Kov โ ๐ฅ๐ฆ | , Kov ๐ฅ๐ฆ < , Kov ๐ฅ๐ฆ Kov + ๐ฅ๐ฆ , Kov ๐ฅ๐ฆ (cid:62) , (9.3)where Kov ๐ฅ๐ฆ = P ( ๐ฅ โฉ ๐ฆ ) โ P ( ๐ฅ ) P ( ๐ฆ ) (9.4)is a covariance of events ๐ฅ and ๐ฆ , and Kov โ ๐ฅ๐ฆ = max { , ๐ ๐ฅ + ๐ ๐ฆ โ } โ ๐ ๐ฅ ๐ ๐ฆ = โ ๐ ๐ฅ ๐ ๐ฆ , Kov + ๐ฅ๐ฆ = min { ๐ ๐ฅ , ๐ ๐ฆ } โ ๐ ๐ฅ ๐ ๐ฆ = ๐ ๐ฆ โ ๐ ๐ฅ ๐ ๐ฆ (9.5)are its the lower and upper Frยดechet-boundaries.From De๏ฌnition (9.3) we get the parametrizationof functional parameter ๐ ๐ โฒ by the double Frยดechet-correlation on the following form: ๐ ๐ โฒ ( ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง ) == {๏ธ ๐ ๐ฅ ๐ ๐ฆ โ Kor ๐ฅ๐ฆ Kov โ ๐ฅ๐ฆ , Kor ๐ฅ๐ฆ < ,๐ ๐ฅ ๐ ๐ฆ + Kor ๐ฅ๐ฆ Kov + ๐ฅ๐ฆ , Kor ๐ฅ๐ฆ (cid:62) {๏ธ ๐ ๐ฅ ๐ ๐ฆ + Kor ๐ฅ๐ฆ ๐ ๐ฅ ๐ ๐ฆ , Kor ๐ฅ๐ฆ < ,๐ ๐ฅ ๐ ๐ฆ + Kor ๐ฅ๐ฆ ( ๐ ๐ฆ โ ๐ ๐ฅ ๐ ๐ฆ ) , Kor ๐ฅ๐ฆ (cid:62) . (9.6) The parametrization of functional parameter ๐ ๐ก โฒ bythe double Frยดechet-correlation is similar: ๐ ๐ก โฒ ( ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง ) == {๏ธ ๐ ๐ฅ ๐ ๐ง โ Kor ๐ฅ๐ง Kov โ ๐ฅ๐ง , Kor ๐ฅ๐ง < ,๐ ๐ฅ ๐ ๐ง + Kor ๐ฅ๐ง Kov + ๐ฅ๐ง , Kor ๐ฅ๐ง (cid:62) {๏ธ ๐ ๐ฅ ๐ ๐ง + Kor ๐ฅ๐ง ๐ ๐ฅ ๐ ๐ง , Kor ๐ฅ๐ง < ,๐ ๐ฅ ๐ ๐ง + Kor ๐ฅ๐ง ( ๐ ๐ง โ ๐ ๐ฅ ๐ ๐ง ) , Kor ๐ฅ๐ง (cid:62) . (9.7) We recall ๏ฌrst that an absolute triple Frยดechet-correlation [1] of three events ๐ฅ, ๐ฆ and ๐ง is de๏ฌnedsimilarly to the double one: Kor ๐ฅ๐ฆ๐ง = โงโชโจโชโฉ Kov ๐ฅ๐ฆ๐ง | Kov โ ๐ฅ๐ฆ๐ง | , Kov ๐ฅ๐ฆ๐ง < , Kov ๐ฅ๐ฆ๐ง
Kov + ๐ฅ๐ฆ๐ง , Kov ๐ฅ๐ฆ๐ง (cid:62) , (9.8)where Kov ๐ฅ๐ฆ๐ง = P ( ๐ฅ โฉ ๐ฆ โฉ ๐ง ) โ P ( ๐ฅ ) P ( ๐ฆ ) P ( ๐ง ) (9.9)is the triple covariance of events ๐ฅ, ๐ฆ and ๐ง , and Kov โ ๐ฅ๐ฆ๐ง = max { , ๐ ๐ฅ + ๐ ๐ฆ + ๐ ๐ง โ } โ ๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง == โ ๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง , Kov + ๐ฅ๐ฆ๐ง = min { ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง } โ ๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง == ๐ ๐ง โ ๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง (9.10)are its absolute the lower and upper Frยดechet-boundaries.The de๏ฌnition of the inserted triple Frยดechet-correlation differs of the de๏ฌnition of absolute one (9.8) in that its the lower and upper Frยดechet-boundaries must depend on the e.p.d. of the in-serted doublets ๐ณ โฒ and ๐ณ โฒโฒ . So they differ from ab-solute Frยดechet-boundaries (9.10) and have the form(9.19), where ๐ โ ๐ โฒ ๐ก โฒ = max { , ๐ ๐ โฒ + ๐ ๐ก โฒ โ ๐ ๐ฅ } ,๐ + ๐ โฒ ๐ก โฒ = min { ๐ ๐ โฒ , ๐ ๐ก โฒ } ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ = max { , ๐ ๐ฅ + ๐ ๐ฆ + ๐ ๐ง โ โ ๐ ๐ โฒ โ ๐ ๐ก โฒ } ,๐ + ๐ โฒโฒ ๐ก โฒโฒ = min { ๐ ๐ฆ โ ๐ ๐ โฒ , ๐ ๐ง โ ๐ ๐ก โฒ } , (9.11)are the lower and upper Frยดechet-boundaries ofprobabilities of double intersections of events from inserted doublets ๐ณ โฒ and ๐ณ โฒโฒ with respect to theframe monoplet { ๐ฅ } , which should serve inserted the lower and upper Frยดechet-boundaries of proba-bilities of triple intersections (9.20) of events fromthe triplet X = { ๐ฅ } + ( ๐ณ โฒ (+) ๐ณ โฒโฒ ) . However, asmight be expected, these Frยดechet-boundaries arenot always ready to serve as the lower and upperFrยดechet-boundaries for probabilities of triple inter-sections.
08 T HE XIV FAMEMSโ2015 C
ONFERENCE
For this reason, it is necessary to modify the de๏ฌ-nitions of two inserted triple covariances and, re-spectively, โ inserted the lower and upper Frยดechet-boundaries of these covariances.
The ๏ฌrst modi๏ฌcation of de๏ฌnitions (see Fig. 18,19,20).
For brevity, we denote ๐ ( { ๐ฅ } ) โ = ๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง , ๐ ( โ ) โ = (1 โ ๐ ๐ฅ ) ๐ ๐ฆ ๐ ๐ง . Two inserted triple covariance are de๏ฌnedby the ๏ฌrmulas: Kov ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง = โงโชโจโชโฉ ๐ ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } ) โ , ๐ ( { ๐ฅ } ) โ โ [๏ธ ๐ โ ๐ โฒ ๐ก โฒ , ๐ + ๐ โฒ ๐ก โฒ ]๏ธ ,๐ โ ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } ) โ , ๐ ( { ๐ฅ } ) โ < ๐ โ ๐ โฒ ๐ก โฒ ,๐ + ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } ) โ , ๐ + ๐ โฒ ๐ก โฒ < ๐ ( { ๐ฅ } ) โ , Kov ( โ ) ๐ฅ๐ฆ๐ง = โงโชโจโชโฉ ๐ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ ) โ , ๐ ( โ ) โ โ [๏ธ ๐ โ ๐ โฒ ๐ก โฒ , ๐ + ๐ โฒ ๐ก โฒ ]๏ธ ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ ) โ , ๐ ( โ ) โ < ๐ โ ๐ โฒโฒ ๐ก โฒโฒ ,๐ + ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ ) โ , ๐ + ๐ โฒโฒ ๐ก โฒโฒ < ๐ ( โ ) โ , (9.12)and inserted the lower and upper Frยดechet-boundaries of these covariances โ by the for-mulas: Kov โ ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง = โงโชโจโชโฉ ๐ โ ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } ) โ , ๐ ( { ๐ฅ } ) โ โ [๏ธ ๐ โ ๐ โฒ ๐ก โฒ , ๐ + ๐ โฒ ๐ก โฒ ]๏ธ ,๐ โ ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } ) โ , ๐ ( { ๐ฅ } ) โ < ๐ โ ๐ โฒ ๐ก โฒ ,๐ + ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } ) โ , ๐ + ๐ โฒ ๐ก โฒ < ๐ ( { ๐ฅ } ) โ , Kov +( { ๐ฅ } ) ๐ฅ๐ฆ๐ง = โงโชโจโชโฉ ๐ + ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } ) โ , ๐ ( { ๐ฅ } ) โ โ [๏ธ ๐ โ ๐ โฒ ๐ก โฒ , ๐ + ๐ โฒ ๐ก โฒ ]๏ธ ,๐ โ ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } ) โ , ๐ ( { ๐ฅ } ) โ < ๐ โ ๐ โฒ ๐ก โฒ ,๐ + ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } ) โ , ๐ + ๐ โฒ ๐ก โฒ < ๐ ( { ๐ฅ } ) โ , (9.13) Kov โ ( โ ) ๐ฅ๐ฆ๐ง = โงโชโจโชโฉ ๐ โ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ ) โ , ๐ ( โ ) โ โ [๏ธ ๐ โ ๐ โฒ ๐ก โฒ , ๐ + ๐ โฒ ๐ก โฒ ]๏ธ ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ ) โ , ๐ ( โ ) โ < ๐ โ ๐ โฒโฒ ๐ก โฒโฒ ,๐ + ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ ) โ , ๐ + ๐ โฒโฒ ๐ก โฒโฒ < ๐ ( โ ) โ , Kov +( โ ) ๐ฅ๐ฆ๐ง = โงโชโจโชโฉ ๐ + ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ ) โ , ๐ ( โ ) โ โ [๏ธ ๐ โ ๐ โฒ ๐ก โฒ , ๐ + ๐ โฒ ๐ก โฒ ]๏ธ ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ ) โ , ๐ ( โ ) โ < ๐ โ ๐ โฒโฒ ๐ก โฒโฒ ,๐ + ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ ) โ , ๐ + ๐ โฒโฒ ๐ก โฒโฒ < ๐ ( โ ) โ , (9.14)We introduce some notation for brevity of the for-mulas: ๐ ( { ๐ฅ } )0 = โงโชโชโชโชโชโจโชโชโชโชโชโฉ ๐ { ๐ฅ } โ , if ๐ { ๐ฅ } โ โ [๏ธ ๐ + ๐ โฒ ๐ก โฒ , ๐ โ ๐ โฒ ๐ก โฒ ]๏ธ ,๐ โ ๐ โฒ ๐ก โฒ + ( ๐ โ ๐ โฒ ๐ก โฒ โ ๐ { ๐ฅ } โ )( ๐ + ๐ โฒ ๐ก โฒ โ ๐ โ ๐ โฒ ๐ก โฒ ) ๐ + ๐ โฒ ๐ก โฒ + ๐ โ ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } ) โ , if ๐ { ๐ฅ } โ ฬธโ [๏ธ ๐ + ๐ โฒ ๐ก โฒ , ๐ โ ๐ โฒ ๐ก โฒ ]๏ธ ,๐ ( โ )0 = โงโชโชโชโชโชโจโชโชโชโชโชโฉ ๐ โ โ , ๐ โ โ โ [๏ธ ๐ + ๐ โฒ ๐ก โฒ , ๐ โ ๐ โฒ ๐ก โฒ ]๏ธ ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ + ( ๐ โ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ ) โ )( ๐ + ๐ โฒ ๐ก โฒ โ ๐ โ ๐ โฒ ๐ก โฒ ) ๐ + ๐ โฒโฒ ๐ก โฒโฒ + ๐ โ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ ) โ ,๐ โ โ ฬธโ [๏ธ ๐ + ๐ โฒ ๐ก โฒ , ๐ โ ๐ โฒ ๐ก โฒ ]๏ธ , (9.15) Two inserted triple covariances are de๏ฌned by theformulas: Kov ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง = ๐ ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } )0 Kov ( โ ) ๐ฅ๐ฆ๐ง = ๐ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ )0 , (9.16)and inserted the lower and upper Frยดechet-boundaries of these covariances โ by the for-mulas: Kov โ ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง = ๐ โ ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } )0 , Kov +( { ๐ฅ } ) ๐ฅ๐ฆ๐ง = ๐ + ๐ โฒ ๐ก โฒ โ ๐ ( { ๐ฅ } )0 , Kov โ ( โ ) ๐ฅ๐ฆ๐ง = ๐ โ ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ )0 , Kov +( โ ) ๐ฅ๐ฆ๐ง = ๐ + ๐ โฒโฒ ๐ก โฒโฒ โ ๐ ( โ )0 . (9.17) โ โ โ For any modi๏ฌcation de๏ฌnitions of two inserted
Frยดechet-correlations look in the usual way:
Kor ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง = โงโชโชโจโชโชโฉ Kov ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง โโโ Kov โ ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง โโโ , Kov ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง < , Kov ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง Kov +( { ๐ฅ } ) ๐ฅ๐ฆ๐ง , Kov ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง (cid:62) , Kor ( โ ) ๐ฅ๐ฆ๐ง = โงโชโชโจโชโชโฉ Kov ( โ ) ๐ฅ๐ฆ๐ง โโโ Kov โ ( โ ) ๐ฅ๐ฆ๐ง โโโ , Kov ( โ ) ๐ฅ๐ฆ๐ง < , Kov ( โ ) ๐ฅ๐ฆ๐ง Kov +( โ ) ๐ฅ๐ฆ๐ง , Kov ( โ ) ๐ฅ๐ฆ๐ง (cid:62) . (9.18) ๐ ๐ โฒ ๐ก โฒ and ๐ ๐ โฒโฒ ๐ก โฒโฒ The Frยดechet-restriction of two functional parame-ters ๐ ๐ โฒ ๐ก โฒ = ๐ ๐ โฒ ๐ก โฒ ( ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง ) ๐ ๐ โฒโฒ ๐ก โฒโฒ = ๐ ๐ โฒโฒ ๐ก โฒโฒ ( ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง ) ๐ โ ๐ โฒ ๐ก โฒ (cid:54) ๐ ๐ โฒ ๐ก โฒ (cid:54) ๐ + ๐ โฒ ๐ก โฒ ,๐ โ ๐ โฒโฒ ๐ก โฒโฒ (cid:54) ๐ ๐ โฒโฒ ๐ก โฒโฒ (cid:54) ๐ + ๐ โฒโฒ ๐ก โฒโฒ , (9.19)that in the frame method have a sense of probabil-ities of triple intersections of events: ๐ ๐ โฒ ๐ก โฒ = P ( ๐ฅ โฉ ๐ฆ โฉ ๐ง ) ,๐ ๐ โฒโฒ ๐ก โฒโฒ = P ( ๐ฅ ๐ โฉ ๐ฆ โฉ ๐ง ) , (9.20)is based on the notion of the inserted triple Frยดechet-correlation.From de๏ฌnitions (9.18) and (9.11) we get theparametrization of functional parameter ๐ ๐ โฒ ๐ก โฒ of the inserted triple Frยดechet-correlation Kor ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง in thefollowing form: ๐ ๐ โฒ ๐ก โฒ ( ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง ) == โงโชโชโชโชโจโชโชโชโชโฉ ๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง โ Kor ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง Kov โ ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง , Kor ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง < ,๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง + Kor ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง Kov +( { ๐ฅ } ) ๐ฅ๐ฆ๐ง , Kor ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง (cid:62) , (9.21) OROBYEV
The parametrization of functional parameter ๐ ๐ โฒโฒ ๐ก โฒโฒ of the inserted triple Frยดechet-correlation Kor ( โ ) ๐ฅ๐ฆ๐ง fol-lows from the same de๏ฌnitions (9.18) and (9.11): ๐ ๐ โฒโฒ ๐ก โฒโฒ ( ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง ) == โงโชโชโชโชโจโชโชโชโชโฉ ๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง โ Kor ( โ ) ๐ฅ๐ฆ๐ง Kov โ ( โ ) ๐ฅ๐ฆ๐ง , Kor ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง < ,๐ ๐ฅ ๐ ๐ฆ ๐ ๐ง + Kor ( โ ) ๐ฅ๐ฆ๐ง Kov +( โ ) ๐ฅ๐ฆ๐ง , Kor ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง (cid:62) , (9.22) Note 20 (about parametrization of functional pa-rameters of 3-Kopula by Frยดechet-correlations).
The parametrization of the four functional pa-rameters ๐ ๐ โฒ , ๐ ๐ก โฒ , ๐ ๐ โฒ ๐ก โฒ and ๐ ๐ โฒโฒ ๐ก โฒโฒ of 3-Kopula of the ห ๐ -ordered half-rare triplet X = { ๐ฅ, ๐ฆ, ๐ง } by twodouble Frยดechet-correlations Kor ๐ฅ๐ฆ and Kor ๐ฅ๐ง (9.3)and by two inserted triple Frยดechet-correlations Kor ( { ๐ฅ } ) ๐ฅ๐ฆ๐ง and Kor ( โ ) ๐ฅ๐ฆ๐ง (9.18) has the following ad-vantages. Each of four Frยดechet-correlations isa numerical characteristics of dependency ofevents with values from ๏ฌxed interval [ โ , +1] .And these values clearly indicate the proxim-ity to Frยดechet-boundaries and to indepedent3-Kopula. The value โ โ โ indicates to the lowerFrยดechet-boundary, the value โ +1 โ โ to the upperFrยดechet-boundary, and the value โ โ โ to inde-pendent events. For example, the equality of allthese four Frยดechet-correlations to zero determinesa family of independent 3-Kopulas. Advantages ofthe proposed idea of parametrization of functionalparameters of 3-Kopula are that โ an each Frยดechet-correlation can take arbitraryvalue from [ โ , +1] without any connectionwith the values of the other three Frยดechet-correlations; โ the above parametrization algorithm for func-tional parameters of 3-Kopula extends to theparametrization of the functional parametersof ๐ -Kopulas by inserted Frยดechet-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 = { ๐ฅ, ๐ฆ } and its set-phenomena X ( ๐ | ๐ฅ ) = { ๐ฅ, ๐ฆ ๐ } , X ( ๐ | ๐ฆ ) = { ๐ฅ ๐ , ๐ฆ } , and X ( ๐ | ๐ฅ๐ฆ ) = { ๐ฅ ๐ , ๐ฆ ๐ } , each of which is char-acterized by its own functional parameter P ( ๐ฅ โฉ In each ๏ฌgure, 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. ๐ฆ ) = ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) , lying within the Frยดechet bound-aries: (cid:54) ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) (cid:54) min { ๐ค ๐ฅ , ๐ค ๐ฆ } . (10.1)Upper 2-Kopula of Frยดechet (embedded): ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) = min { ๐ค ๐ฅ , ๐ค ๐ฆ } . (10.2)Independent 2-Kopula of Frยดechet: ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) = ๐ค ๐ฅ ๐ค ๐ฆ . (10.3)Lower 2-Kopula of Frยดechet (minimum-intersected): ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) = max { , ๐ค ๐ฅ + ๐ค ๐ฆ โ } . (10.4)Half-independent 2-Kopula: ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) = ๐ค ๐ฅ ๐ค ๐ฆ / . (10.5)Half-embedded 2-Kopula: ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) = min { ๐ค ๐ฅ , ๐ค ๐ฆ } / . (10.6)Arbitrary-embedded 2-Kopula: ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) == min { ๐ค ๐ฅ , ๐ค ๐ฆ } (1 + sin(15( ๐ค ๐ฅ โ ๐ค ๐ฆ ))) / . (10.7)Continuously-arbitrary-embedded 2-Kopula: ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) == ๐ค ๐ฅ ๐ค ๐ฆ + ( ๐ผ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) โ ๐ค ๐ฅ ๐ค ๐ฆ ) ๐ฝ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) , (10.8)where ๐ผ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) == min { ๐ค ๐ฅ , ๐ค ๐ฆ } (1 + sin(15( ๐ค ๐ฅ โ ๐ค ๐ฆ ))) / ,๐ฝ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) = โ๏ธ (1 / โ ๐ค ๐ฅ )(1 / โ ๐ค ๐ฆ ) . (10.9) In Fig.s 13, 14, 16, 17, and 15 it is shown 2-Kopulasof doublets of half-rare events X = { ๐ฅ, ๐ฆ } and its set-phenomena X ( ๐ | ๐ฅ ) , X ( ๐ | ๐ฆ ) , and X ( ๐ | ๐ฅ๐ฆ ) , correspondingto some classical copulas.2-Kopula of Ali-Mikhail-Haq, ๐ โ [ โ , : ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) == ๐ค ๐ฅ ๐ค ๐ฆ โ ๐ (1 โ ๐ค ๐ฅ )(1 โ ๐ค ๐ฆ ) . (10.10)
10 T HE XIV FAMEMSโ2015 C
ONFERENCE
Figure 12: Cartesian representations of 2-Kopulas of doublets of half-rareevents X = { ๐ฅ, ๐ฆ } and its set-phenomena X ( ๐ | ๐ฅ ) , X ( ๐ | ๐ฆ ) X ( ๐ | ๐ฅ๐ฆ ) 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 = { ๐ฅ, ๐ฆ } and its set-phenomena X ( ๐ | ๐ฅ ) , X ( ๐ | ๐ฆ ) X ( ๐ | ๐ฅ๐ฆ ) corre-sponding to Ali-Mikhail-Haq Kopula (from up to down): from near-upper( ๐ = 0 . ) through independent ( ๐ = 0 ) to lower ( ๐ = โ . ). ๐ โ [ โ , โ ) โ { } : ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) == [๏ธ max {๏ธ ๐ค โ ๐๐ฅ + ๐ค โ ๐๐ฆ โ
1; 0 }๏ธ]๏ธ โ /๐ . (10.11) Figure 14: Cartesian representations of 2-Kopulas of doublets of half-rareevents X = { ๐ฅ, ๐ฆ } and its set-phenomena X ( ๐ | ๐ฅ ) , X ( ๐ | ๐ฆ ) X ( ๐ | ๐ฅ๐ฆ ) corre-sponding to Clayton Kopula (from up to down): from near-upper ( ๐ =6 . ) around pinked-independent ( ๐ = 0 . , โ . ) to lower ( ๐ = โ . ).Figure 15: Cartesian representations of 2-Kopulas of doublets of half-rareevents X = { ๐ฅ, ๐ฆ } and its set-phenomena X ( ๐ | ๐ฅ ) , X ( ๐ | ๐ฆ ) X ( ๐ | ๐ฅ๐ฆ ) corre-sponding to Joe Kopula (from up to down): from independent ( ๐ = 1 . )to near-lower ( ๐ = 6 . ). ๐ โ [1 , โ ) : ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) == 1 โ [๏ธ (1 โ ๐ค ๐ฅ ) ๐ +(1 โ ๐ค ๐ฆ ) ๐ โ (1 โ ๐ค ๐ฅ ) ๐ (1 โ ๐ค ๐ฆ ) ๐ ]๏ธ /๐ . (10.12) OROBYEV X = { ๐ฅ, ๐ฆ } and its set-phenomena X ( ๐ | ๐ฅ ) , X ( ๐ | ๐ฆ ) X ( ๐ | ๐ฅ๐ฆ ) corre-sponding to Frank Kopula (from up to down): from near-upper ( ๐ = 6 . )around pinked-independent ( ๐ = 0 . , โ . ) to near-lower ( ๐ = โ . ).Figure 17: Cartesian representations of 2-Kopulas of doublets of half-rareevents X = { ๐ฅ, ๐ฆ } and its set-phenomena X ( ๐ | ๐ฅ ) , X ( ๐ | ๐ฆ ) X ( ๐ | ๐ฅ๐ฆ ) cor-responding to Gumbel Kopula (from up to down): from inndependent( ๐ = 1 . ) to near-lower ( ๐ = 6 . ). ๐ โ R โ { } : ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) == โ ๐ log [๏ธ โ ๐๐ค ๐ฅ ) โ โ ๐๐ค ๐ฆ ) โ โ ๐ ) โ ]๏ธ . (10.13) ๐ โ [1 , โ ) : ๐ ๐ฅ๐ฆ ( ๐ค ๐ฅ , ๐ค ๐ฆ ) == exp [๏ธ โ (๏ธ ( โ log( ๐ค ๐ฅ )) ๐ + ( โ log( ๐ค ๐ฆ )) ๐ )๏ธ /๐ ]๏ธ . (10.14) In Fig.โs
18, 19, and 20 it is shown 3-Kopulas oftriplets of half-rare events X = { ๐ฅ, ๐ฆ, ๐ง } , functionalparameters of which serve Frยดechet-correlations inthe ๏ฌrst modi๏ฌcation of de๏ฌnitions (see paragraph9.2).
11 Appendix
Consider the universal probability space (โฆ , ๐ (cid:102) , P ) and one of its subject-name realizations , a par-tial probability space (โฆ , ๐ , P ) . The elements ofthe sigma-algebra ๐ (cid:102) are the universal Kolmogorovevents ๐ฅ (cid:102) โ ๐ (cid:102) , and the elements of the sigma-algebra ๐ โ events ๐ฅ โ ๐ , which serve as names ofuniversal Kolmogorov events ๐ฅ (cid:102) (see in details [8]).The notions that are relevant to a s.e. X โ ๐ forwhich it is convenient to use the following abbrevi-ations: X = { ๐ฅ : ๐ฅ โ X } โ a set of events (s.e.); ห ๐ = { ๐ ๐ฅ , ๐ฅ โ X } โ an X -set of probabilities of events from X ; X ( ๐ | ๐ ) = X ( ๐ | ๐// X ) = { ๐ฅ : ๐ฅ โ ๐ } + { ๐ฅ ๐ : ๐ฅ โ X โ ๐ } โ an ๐ -phenomenon of X , ๐ โ X ; X ( ๐ | X ) = X ( ๐ | X // X ) = X โ an X -phenomenon of X equal to X ; ห ๐ ( ๐ | ๐// X ) = { ๐ ๐ฅ , ๐ฅ โ ๐ } + { โ ๐ ๐ฅ , ๐ฅ โ X โ ๐ } โ an X -set of probabilities of events from X ( ๐ | ๐ ) , ๐ โ X ; ห ๐ ( ๐ | X // X ) = ห ๐ โ an X -set of probabilities of events from X ( ๐ | X ) equal to ห ๐ ; ๐ ( ๐// X ) โ a value of e.p.d. of the 1st kind of X for ๐ โ X ; K (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ โ a value of the Kopula of e.p.d. of the 1st kind of X for ๐ โ X ; ๐ ( ๐// X ) = K (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ โ the de๏ฌnition of e.p.d. of the 1st kind of X by its Kopula, ๐ โ X ; ห ๐ = { ๐ ๐ฅ , ๐ฅ โ X } โ [0 , / โ X โ an X -set of half-rare variables; ห ๐ค = { ๐ค ๐ฅ , ๐ฅ โ X } โ [0 , โ X โ an X -set of free variables. Each ๏ฌgure 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.
12 T HE XIV FAMEMSโ2015 C
ONFERENCE
Kor = โ Kor = โ . Kor = โ . Kor = โ . Kor = 0
Kor = 0 . Kor = 0 . Kor = 0 . Kor = 1
Figure 18: The ๏ฌrst modi๏ฌcation of de๏ฌnitions. Cartesian representationsof 3-Kopulas of triplets of half-rare events X = { ๐ฅ, ๐ฆ, ๐ง } , constructed bythe frame method (6.18) with non-negative values of a single parame-ter (from up to down) Kor = โ , โ . , โ . , โ . , , . , . , . , , 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 ๐ -phenomena X ( ๐ | ๐ ) are connected by formulas of mutually inversionset-phenomenon renumbering for ๐ โ X ๐ โ X : ๐ (๏ธ ๐ ( ๐ | ๐ โฉ ๐ ) // X ( ๐ | ๐ ) )๏ธ = ๐ (( ๐ โ ๐ ) ๐ // X ) ,๐ ( ๐// X ) = ๐ (๏ธ (( ๐ โ ๐ ) ๐ ) ( ๐ | ๐ โฉ ( ๐ ฮ ๐ ) ๐ ) // X ( ๐ | ๐ ) )๏ธ ; (11.1) Kor = โ Kor = โ . Kor = โ . Kor = โ . Kor = โ . Kor = โ . Kor = โ . Kor = โ . Kor = โ . Kor = โ . Kor = 0
Figure 19: The ๏ฌrst modi๏ฌcation of de๏ฌnitions. Cartesian representationsof 3-Kopulas of triplets of half-rare events X = { ๐ฅ, ๐ฆ, ๐ง } , constructed bythe frame method (6.18) with non-positive values of the parameter Kor = โ , โ . , ..., โ . , โ . , (from up to down), to which inserted Frechet-correlations are equal (see paragraph 9). The independent 3-Kopula isobtained for
Kor = 0 . K (๏ธ ห ๐ ( ๐ | ๐ ( ๐ | ๐ โฉ ๐ ) // X ( ๐ | ๐ ) ) )๏ธ = K (๏ธ ห ๐ ( ๐ | ( ๐ ฮ ๐ ) ๐ // X ) )๏ธ , K (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ = K (๏ธ ห ๐ ( ๐ | (( ๐ ฮ ๐ ) ๐ ) ( ๐ | ๐ โฉ ( ๐ ฮ ๐ ) ๐ ) // X ( ๐ | ๐ ) ) )๏ธ . (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 ๏ฌrst modi๏ฌcation of de๏ฌnitions. Cartesian representationsof 3-Kopulas of triplets of half-rare events X = { ๐ฅ, ๐ฆ, ๐ง } , 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]. ห ๐ -order The following special denotations for a doublet ofevents X = { ๐ฅ, ๐ฆ } and the X -set of marginal proba-bilities ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ } that are invariant relative to the ห ๐ -order, are useful. X = { ๐ฅ, ๐ฆ } = { ๐ฅ โ , ๐ฅ โ } , ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ } = { ๐ โ , ๐ โ } , / (cid:62) ๐ โ (cid:62) ๐ โ , ห ๐ค = { ๐ค ๐ฅ , ๐ค ๐ฆ } = { ๐ค โ , ๐ค โ } , (cid:62) ๐ค โ (cid:62) ๐ค โ , (11.3) { ๐ฅ โ } = max { X } = {๏ธ { ๐ฅ } , ๐ ๐ฅ > ๐ ๐ฆ , { ๐ฆ } , ; { ๐ฅ โ } = min { X } = {๏ธ { ๐ฅ } , ๐ ๐ฅ (cid:54) ๐ ๐ฆ , { ๐ฆ } , ; (11.4) ๐ค โ = max { ห ๐ค } = {๏ธ ๐ค ๐ฅ , ๐ค ๐ฅ > ๐ค ๐ฆ ,๐ค ๐ฆ , ; ๐ค โ = min { ห ๐ค } = {๏ธ ๐ค ๐ฅ , ๐ค ๐ฅ (cid:54) ๐ค ๐ฆ ,๐ค ๐ฆ , ; (11.5) ๐ โ = max { ห ๐ } = {๏ธ ๐ ๐ฅ , ๐ ๐ฅ > ๐ ๐ฆ ,๐ ๐ฆ , ;= max {๏ธ min { ๐ค ๐ฅ , โ ๐ค ๐ฅ } , min { ๐ค ๐ฆ , โ ๐ค ๐ฆ } }๏ธ ,๐ โ = min { ห ๐ } = {๏ธ ๐ ๐ฅ , ๐ ๐ฅ (cid:54) ๐ ๐ฆ ,๐ ๐ฆ , ;= min {๏ธ min { ๐ค ๐ฅ , โ ๐ค ๐ฅ } , min { ๐ค ๐ฆ , โ ๐ค ๐ฆ } }๏ธ . (11.6)In such invariant denotations, it is not di๏ฌcult towrite down the general recurrence formula for thehalf-rare 2-Kopula of the doublet X , united combin-ing both orders: K (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ = K ( ๐ | X // X ) (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ == โงโชโชโชโจโชโชโชโฉ K โฒ ( ๐ ๐ฅ๐ฆ (ห ๐ )) , ๐ = X , K โฒโฒ (๏ธ ๐ โ โ ๐ ๐ฅ๐ฆ (ห ๐ ) )๏ธ , ๐ = { ๐ฅ โ } , K โฒ (1 โ ๐ ๐ฅ๐ฆ (ห ๐ )) โ ๐ โ , ๐ = { ๐ฅ โ } , K โฒโฒ (๏ธ โ ๐ โ + ๐ ๐ฅ๐ฆ (ห ๐ ) )๏ธ โ ๐ โ , ๐ = โ (11.7)where by De๏ฌnition (11.3) ๐ โ = max {๏ธ min { ๐ค ๐ฅ , โ ๐ค ๐ฅ } , min { ๐ค ๐ฆ , โ ๐ค ๐ฆ } }๏ธ ,๐ โ = min {๏ธ min { ๐ค ๐ฅ , โ ๐ค ๐ฅ } , min { ๐ค ๐ฆ , โ ๐ค ๐ฆ } }๏ธ .
14 T HE XIV FAMEMSโ2015 C
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The mutual set-phenomenon inversion of 2-Kopulas of half-rare ห ๐ and free ห ๐ค marginal proba-bilities has the form: K (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ = K ( ๐ | ๐// X ) ( ห ๐ค ) , K ( ๐ | ๐// X ) (ห ๐ ) = K (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ ; (11.8) K ( ๐ | ๐// X ) (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ = K ( ๐ | X // X ) (๏ธ ห ๐ ( ๐ | ( ๐ ๐ ฮ ๐ ) ๐ // X ) )๏ธ = K (๏ธ ห ๐ ( ๐ | ( ๐ ๐ ฮ ๐ ) ๐ // X ) )๏ธ . (11.9)For example, for ๐ โ X = { ๐ฅ, ๐ฆ } K ( ๐ | X // X ) (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ = K (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ , K ( ๐ |{ ๐ฅ } // X ) (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ = K (๏ธ ห ๐ ( ๐ | ( { ๐ฆ } ฮ ๐ ) ๐ // X ) )๏ธ , K ( ๐ |{ ๐ฆ } // X ) (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ = K (๏ธ ห ๐ ( ๐ | ( { ๐ฅ } ฮ ๐ ) ๐ // X ) )๏ธ , K ( ๐ |โ // X ) (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ = K (๏ธ ห ๐ ( ๐ | ๐ ๐ // X ) )๏ธ . (11.10) ห ๐ -ordering ahalf-rare s.e. ห ๐ -ordering ahalf-rare doublets of events Let us explain the role of ห ๐ -ordering in the framemethod using the example of constructing a 2-Kopula of the ห ๐ -ordered half-rare events X = { ๐ฅ, ๐ฆ } with X -set of marginal probabilities of events ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ } , that is, / (cid:62) ๐ ๐ฅ (cid:62) ๐ ๐ฆ : K โฒ (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ } ) )๏ธ = โงโชโชโชโจโชโชโชโฉ ๐ ๐ฅ๐ฆ , ๐ = { ๐ฅ, ๐ฆ } ,๐ ๐ฅ โ ๐ ๐ฅ๐ฆ , ๐ = { ๐ฅ } ,๐ ๐ฆ โ ๐ ๐ฅ๐ฆ , ๐ = { ๐ฆ } , โ ๐ ๐ฅ โ ๐ ๐ฆ + ๐ ๐ฅ๐ฆ , ๐ = โ , (11.11)where, when selected as a function parameter ๐ ๐ฅ๐ฆ of the 1-Kopulas of inserted half-rare monoplates ๐ณ โฒ = { ๐ โฒ } = { ๐ฅ โฉ ๐ฆ } and ๐ณ โฒโฒ = { ๐ โฒโฒ } = { ๐ฅ ๐ โฉ ๐ฆ } , areequal, respectively: K โฒ (๏ธ ห ๐ ( ๐ | ๐// ๐ณ โฒ ) )๏ธ = โงโชโชโชโจโชโชโชโฉ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ โฒ , ๐ก โฒ } ,๐ ๐ โฒ โ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ โฒ } ,๐ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ก โฒ } , โ ๐ ๐ โฒ โ ๐ ๐ก โฒ + ๐ ๐ โฒ ๐ก โฒ , ๐ = โ , K โฒโฒ (๏ธ ห ๐ ( ๐ | ๐// ๐ณ โฒโฒ ) )๏ธ == โงโชโชโชโจโชโชโชโฉ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ โฒโฒ , ๐ก โฒโฒ } ,๐ ๐ฆ โ ๐ ๐ โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ โฒโฒ } ,๐ ๐ง โ ๐ ๐ก โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ก โฒโฒ } , โ ๐ ๐ฆ โ ๐ ๐ง + ๐ ๐ โฒ + ๐ ๐ก โฒ + ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = โ , (11.12) under the assumption that inserted half-raremonoplets have โequally directโ ห ๐ -orders: ๐ ๐ฆ (cid:62) ๐ ๐ โฒ (cid:62) ๐ ๐ก โฒ ,๐ ๐ง (cid:62) ๐ ๐ โฒโฒ (cid:62) ๐ ๐ก โฒโฒ . (11.13)However, nothing prevents the emergence of twomore โopposite ห ๐ ordersโ on the inserted half-raremonoplets: ๐ ๐ฆ (cid:62) ๐ ๐ก โฒ (cid:62) ๐ ๐ โฒ ,๐ ๐ง (cid:62) ๐ ๐ โฒโฒ (cid:62) ๐ ๐ก โฒโฒ . (11.14) ๐ ๐ฆ (cid:62) ๐ ๐ โฒ (cid:62) ๐ ๐ก โฒ ,๐ ๐ง (cid:62) ๐ ๐ก โฒโฒ (cid:62) ๐ ๐ โฒโฒ ; (11.15)except for the โequally inverseโ ห ๐ -order ๐ ๐ฆ (cid:62) ๐ ๐ก โฒ (cid:62) ๐ ๐ โฒ ,๐ ๐ง (cid:62) ๐ ๐ก โฒโฒ (cid:62) ๐ ๐ โฒโฒ , (11.16)which can not be due to the consistency of the func-tional parameters, i.e., because ๐ ๐ฆ = ๐ ๐ โฒ + ๐ ๐ โฒโฒ (cid:62) ๐ ๐ก โฒ + ๐ ๐ก โฒโฒ = ๐ ๐ง . (11.17) ห ๐ -ordering thehalf-rare triplets of events Let us explain the role of ห ๐ -ordering in the framemethod using the example of constructing a 3-Kopula of the ห ๐ -ordered half-rare events X = { ๐ฅ, ๐ฆ, ๐ง } with X -set of marginal probabilities ofevents ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง } , that is, / (cid:62) ๐ ๐ฅ (cid:62) ๐ ๐ฆ (cid:62) ๐ ๐ง : K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ,๐ฆ,๐ง } ) )๏ธ == โงโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ K โฒ ( ๐ ๐ โฒ , ๐ ๐ก โฒ ) , ๐ = { ๐ฅ, ๐ฆ, ๐ง } , K โฒ ( ๐ ๐ โฒ , โ ๐ ๐ก โฒ ) , ๐ = { ๐ฅ, ๐ฆ } , K โฒ (1 โ ๐ ๐ โฒ , ๐ ๐ก โฒ ) , ๐ = { ๐ฅ, ๐ง } , K โฒ (1 โ ๐ ๐ โฒ , โ ๐ ๐ก โฒ ) โ ๐ ๐ฅ , ๐ = { ๐ฅ } , K โฒโฒ ( ๐ ๐ฆ โ ๐ ๐ โฒ , ๐ ๐ง โ ๐ ๐ก โฒ ) , ๐ = { ๐ฆ, ๐ง } , K โฒโฒ ( ๐ ๐ฆ โ ๐ ๐ โฒ , โ ๐ ๐ง + ๐ ๐ก โฒ ) , ๐ = { ๐ฆ } , K โฒโฒ (1 โ ๐ ๐ฆ + ๐ ๐ โฒ , ๐ ๐ง โ ๐ ๐ก โฒ ) , ๐ = { ๐ง } , K โฒโฒ (1 โ ๐ ๐ฆ + ๐ ๐ โฒ , โ ๐ ๐ง + ๐ ๐ก โฒ ) โ ๐ ๐ฅ , ๐ = โ , (11.18)where, when selected as function parameters ๐ ๐ โฒ , ๐ ๐ก โฒ , ๐ ๐ โฒ ๐ก โฒ and ๐ ๐ โฒโฒ ๐ก โฒโฒ and despite the fact that ๐ ๐ โฒโฒ = ๐ ๐ฆ โ ๐ ๐ โฒ , ๐ ๐ก โฒโฒ = ๐ ๐ง โ ๐ ๐ก โฒ , the 2-Kopulas of insertedhalf-rare doublets ๐ณ โฒ = { ๐ โฒ , ๐ก โฒ } = { ๐ฅ โฉ ๐ฆ, ๐ฅ โฉ ๐ง } and OROBYEV ๐ณ โฒโฒ = { ๐ โฒโฒ , ๐ก โฒโฒ } = { ๐ฅ ๐ โฉ ๐ฆ, ๐ง ๐ โฉ ๐ง } are equal, respectively: K โฒ (๏ธ ห ๐ ( ๐ | ๐// ๐ณ โฒ ) )๏ธ = โงโชโชโชโจโชโชโชโฉ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ โฒ , ๐ก โฒ } ,๐ ๐ โฒ โ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ โฒ } ,๐ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ก โฒ } , โ ๐ ๐ โฒ โ ๐ ๐ก โฒ + ๐ ๐ โฒ ๐ก โฒ , ๐ = โ , K โฒโฒ (๏ธ ห ๐ ( ๐ | ๐// ๐ณ โฒโฒ ) )๏ธ == โงโชโชโชโจโชโชโชโฉ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ โฒโฒ , ๐ก โฒโฒ } ,๐ ๐ฆ โ ๐ ๐ โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ โฒโฒ } ,๐ ๐ง โ ๐ ๐ก โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ก โฒโฒ } , โ ๐ ๐ฆ โ ๐ ๐ง + ๐ ๐ โฒ + ๐ ๐ก โฒ + ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = โ , (11.19)under the assumption that the inserted half-raredoublets have โequally directโ ห ๐ -orders: ๐ ๐ฆ (cid:62) ๐ ๐ โฒ (cid:62) ๐ ๐ก โฒ ,๐ ๐ง (cid:62) ๐ ๐ โฒโฒ (cid:62) ๐ ๐ก โฒโฒ . (11.20)However, nothing prevents the emergence of twomore โopposite ห ๐ ordersโ on the inserted half-raredoublets: ๐ ๐ฆ (cid:62) ๐ ๐ก โฒ (cid:62) ๐ ๐ โฒ ,๐ ๐ง (cid:62) ๐ ๐ โฒโฒ (cid:62) ๐ ๐ก โฒโฒ . (11.21) ๐ ๐ฆ (cid:62) ๐ ๐ โฒ (cid:62) ๐ ๐ก โฒ ,๐ ๐ง (cid:62) ๐ ๐ก โฒโฒ (cid:62) ๐ ๐ โฒโฒ ; (11.22)except for the โequally inverseโ ห ๐ -order ๐ ๐ฆ (cid:62) ๐ ๐ก โฒ (cid:62) ๐ ๐ โฒ ,๐ ๐ง (cid:62) ๐ ๐ก โฒโฒ (cid:62) ๐ ๐ โฒโฒ , (11.23)which can not be due to the consistency of the func-tional parameters, i.e., because ๐ ๐ฆ = ๐ ๐ โฒ + ๐ ๐ โฒโฒ (cid:62) ๐ ๐ก โฒ + ๐ ๐ก โฒโฒ = ๐ ๐ง . (11.24) ห ๐ -non-ordered half-rare s.e.โs Above we outlined the frame method for construct-ing ๐ -Kopulas of ห ๐ - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered half-rare ๐ -s.e.โs. Itremains to extend it to construct ๐ -Kopulas of ห ๐ - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) disordered half-rare ๐ -s.e.โs using the followingtechnique, based on the obvious invariance prop-erty of permutations of events in s.e.: โas eventsfrom some s.e. do not order, the s.e. will not changeโ; and very useful in practical calculations.We denote by X * = { ๐ฅ * , ๐ฅ * , ..., ๐ฅ * ๐ โ } , (11.25)โ the ห ๐ - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered half-rare ๐ -s.e. , which consistsfrom the same events, that an โarbitraryโ ห ๐ - (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 ๐ -s.e. X = { ๐ฅ , ๐ฅ , ..., ๐ฅ ๐ โ } , (11.26) i.e., X * = { ๐ฅ * , ๐ฅ * , ..., ๐ฅ * ๐ โ } = { ๐ฅ , ๐ฅ , ..., ๐ฅ ๐ โ } = X , (11.27)but arranged in descending order of their probabil-ities. In other words, the X * -set of marginal proba-bilities ห ๐ * = { ๐ * , ๐ * , ..., ๐ * ๐ โ } , (11.28)is such that / (cid:62) ๐ * (cid:62) ๐ * (cid:62) ... (cid:62) ๐ * ๐ โ (11.29)where ๐ * = max { ๐ ๐ฅ : ๐ฅ โ X } ,๐ * = max { ๐ ๐ฅ : ๐ฅ โ X โ { ๐ฅ * }} ,...๐ * ๐ +1 = max { ๐ ๐ฅ : ๐ฅ โ X โ { ๐ฅ * , ..., ๐ฅ * ๐ }} ,...๐ * ๐ = max { ๐ ๐ฅ : ๐ฅ โ X โ { ๐ฅ * , ..., ๐ฅ * ๐ โ }} . (11.30)Consequently, X * -set of marginal probabilities ห ๐ * which consists of the same probabilities that X -setof marginal probabilities ห ๐ , i.e., ห ๐ * = { ๐ * , ๐ * , ..., ๐ * ๐ โ } = { ๐ , ๐ , ..., ๐ ๐ โ } = ห ๐, (11.31)but arranged in descending order.Now, to construct the ๐ -Kopulas of the ห ๐ - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) disordered ๐ -s.e X by the frame method it is suf-๏ฌcient to construct this ๐ -Kopula of the ห ๐ - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered ๐ -s.e X * = X by this method, reasoning by (11.27)and (11.31) reasoning that ๐ - ห ๐ - ๐ -s.e. X ๐ - ห ๐ - ๐ -s.e. X * = X , reasoningby virtue of (11.27) and (11.31), that K (ห ๐ ) = K (ห ๐ * ) , (11.32)i.e., for ๐ โ X K (๏ธ ห ๐ ( ๐ | ๐// X ) )๏ธ = K (๏ธ ห ๐ * ( ๐ | ๐ * // X * ) )๏ธ (11.33)where ๐ * = { ๐ฅ * : ๐ฅ โ ๐ } โ X * (11.34)are subsets of the ห ๐ - (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) ordered ( ๐ โ -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
16 T HE XIV FAMEMSโ2015 C
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., ห ๐ -disordered, s.e.โs X = { ๐ฅ , ๐ฅ , ..., ๐ฅ ๐ โ } in the notation just introduced, assuming that wehave available Kopulas of the ห ๐ -ordered ๐ -s.eโs X * = { ๐ฅ * , ๐ฅ * , ..., ๐ฅ * ๐ โ } = X for ๐ = 1 , . Example 2 (invariant formula for the 2-Kopula of ahalf-rare doublet of events).
Let X = { ๐ฅ , ๐ฅ } bethe ห ๐ -non-ordered half-rare doublet of events.Then its 2-Kopula is calculated at each point ห ๐ ( ๐ | ๐// { ๐ฅ ,๐ฅ } ) โ [0 , โ X by the following formulas: K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ ,๐ฅ } ) )๏ธ = K (๏ธ ห ๐ ( ๐ | ๐ * // { ๐ฅ * ,๐ฅ * } ) )๏ธ == โงโชโชโชโจโชโชโชโฉ K โฒ ( ๐ ๐ โฒ ) , ๐ * = { ๐ฅ * , ๐ฅ * } , K โฒโฒ ( ๐ โ ๐ ๐ โฒ ) , ๐ * = { ๐ฅ * } , K โฒ (1 โ ๐ ๐ โฒ ) โ ๐ , ๐ * = { ๐ฅ * } , K โฒโฒ (1 โ ๐ + ๐ ๐ โฒ ) โ ๐ , ๐ * = โ , (11.35)where ๐ณ โฒ = { ๐ โฒ } = { ๐ฅ * โฉ ๐ฅ * } , ๐ณ โฒโฒ = { ๐ โฒโฒ } = { ( ๐ฅ * ) ๐ โฉ ๐ฅ * } . (11.36) Example 3 (invariant formula for the 3-Kopula of ahalf-rare triplet of events).
Let X = { ๐ฅ , ๐ฅ , ๐ฅ } be the ห ๐ -non-ordered half-rare triplet of events.Then its 3-Kopula is calculated at each point ห ๐ ( ๐ | ๐// { ๐ฅ ,๐ฅ ,๐ฅ } ) โ [0 , โ X by the following formulas: K (๏ธ ห ๐ ( ๐ | ๐// { ๐ฅ ,๐ฅ ,๐ฅ } ) )๏ธ = K (๏ธ ห ๐ ( ๐ | ๐ * // { ๐ฅ * ,๐ฅ * ,๐ฅ * } ) )๏ธ == โงโชโชโชโชโชโชโชโชโชโชโชโชโชโจโชโชโชโชโชโชโชโชโชโชโชโชโชโฉ K โฒ ( ๐ ๐ โฒ , ๐ ๐ก โฒ ) , ๐ * = { ๐ฅ * , ๐ฅ * , ๐ฅ * } , K โฒ ( ๐ ๐ โฒ , โ ๐ ๐ก โฒ ) , ๐ * = { ๐ฅ * , ๐ฅ * } , K โฒ (1 โ ๐ ๐ โฒ , ๐ ๐ก โฒ ) , ๐ * = { ๐ฅ * , ๐ฅ * } , K โฒ (1 โ ๐ ๐ โฒ , โ ๐ ๐ก โฒ ) โ ๐ , ๐ * = { ๐ฅ * } , K โฒโฒ ( ๐ โ ๐ ๐ โฒ , ๐ โ ๐ ๐ก โฒ ) , ๐ * = { ๐ฅ * , ๐ฅ * } , K โฒโฒ (1 โ ๐ + ๐ ๐ โฒ , ๐ โ ๐ ๐ก โฒ ) , ๐ * = { ๐ฅ * } , K โฒโฒ ( ๐ โ ๐ ๐ โฒ , โ ๐ + ๐ ๐ก โฒ ) , ๐ * = { ๐ฅ * } , K โฒโฒ (1 โ ๐ + ๐ ๐ โฒ , โ ๐ + ๐ ๐ก โฒ ) โ ๐ , ๐ * = โ , (11.37)where when selecting as function parameters ๐ ๐ โฒ , ๐ ๐ก โฒ , ๐ ๐ โฒ ๐ก โฒ ๐ ๐ โฒโฒ ๐ก โฒโฒ and despite the fact that ๐ ๐ โฒโฒ = ๐ โ ๐ ๐ โฒ , ๐ ๐ก โฒโฒ = ๐ โ ๐ ๐ก โฒ , 2-Kopulas of the inserted half-rare doublets ๐ณ โฒ = { ๐ โฒ , ๐ก โฒ } = { ๐ฅ * โฉ ๐ฅ * , ๐ฅ * โฉ ๐ฅ * } and ๐ณ โฒโฒ = { ๐ โฒโฒ , ๐ก โฒโฒ } = { ( ๐ฅ * ) ๐ โฉ ๐ฅ * , ( ๐ฅ * ) ๐ โฉ ๐ฅ * } are equal re- spectively: K โฒ (๏ธ ห ๐ ( ๐ | ๐// ๐ณ โฒ ) )๏ธ = โงโชโชโชโจโชโชโชโฉ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ โฒ , ๐ก โฒ } ,๐ ๐ โฒ โ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ โฒ } ,๐ ๐ก โฒ โ ๐ ๐ โฒ ๐ก โฒ , ๐ = { ๐ก โฒ } , โ ๐ ๐ โฒ โ ๐ ๐ก โฒ + ๐ ๐ โฒ ๐ก โฒ , ๐ = โ , K โฒโฒ (๏ธ ห ๐ ( ๐ | ๐// ๐ณ โฒโฒ ) )๏ธ == โงโชโชโชโจโชโชโชโฉ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ โฒโฒ , ๐ก โฒโฒ } ,๐ โ ๐ ๐ โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ โฒโฒ } ,๐ โ ๐ ๐ก โฒ โ ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = { ๐ก โฒโฒ } , โ ๐ โ ๐ + ๐ ๐ โฒ + ๐ ๐ก โฒ + ๐ ๐ โฒโฒ ๐ก โฒโฒ , ๐ = โ . (11.38) For a subset of events ๐ โ X ๐ -phenomenonrenumbering of the terrace events, generated by ( ๐ โ -s.h-r.e. X , is based on the replacement ofevents from the subset ๐ ๐ = X โ ๐ by their comple-ments: X ( ๐ | ๐ ) = ๐ + ( ๐ ๐ ) ( ๐ ) == { ๐ฅ, ๐ฅ โ ๐ } + { ๐ฅ ๐ , ๐ฅ โ ๐ ๐ } , (11.39)from which the mutually inverse set-phenomenonrenumbering formulas follow: ter (๏ธ ๐ ( ๐ | ๐ โฉ ๐ ) // X ( ๐ | ๐ ) )๏ธ = ter (( ๐ โ ๐ ) ๐ // X ) , ter ( ๐// X ) = ter (๏ธ (( ๐ โ ๐ ) ๐ ) ( ๐ | ๐ โฉ ( ๐ ฮ ๐ ) ๐ ) // X ( ๐ | ๐ ) )๏ธ (11.40)for ๐ โ X and ๐ โ X .Therefore, the ๐ -phenomenon renumbering of theterrace events, generated by ( ๐ โ -s.h-r.e. X , by theformulas (11.40) is geometrically interpreted on the ( ๐ โ -dimensional Venn diagram of this s.e. as are๏ฌection of the X -hypercube relative to those hy-perplanes that are orthogonal to the ๐ฅ -axes num-bered by the events ๐ฅ โ ๐ ๐ โ X (see Fig. 21 for thedoublet of events). X -simplex on the X -hypercube Take an arbitrary ( ๐ โ -s.e. X โ ๐ with e.p.d. ofthe 1st kind, which, as is known [1], is de๏ฌned asthe X -set of probabilities of terrace events of the1st kind { ๐ ( ๐// X ) , ๐ โ X } . (11.41)Look at the X -set (11.41) as a X -hyper-point from ahalf-rare ๐ -vertex simplex S X == โงโจโฉ { ๐ ( ๐// X ) , ๐ โ X } : ๐ ( ๐// X ) (cid:62) , โ๏ธ ๐ โ X ๐ ( ๐// X )=1 โซโฌโญ , (11.42) OROBYEV { ๐ฅ, ๐ฆ ๐ } { ๐ฆ ๐ } { ๐ฆ ๐ } { ๐ฅ ๐ , ๐ฆ ๐ } X ( ๐ | ๐ฅ ) X ( ๐ |โ ) { ๐ฅ } โ โ { ๐ฅ ๐ }{ ๐ฅ } โ โ { ๐ฅ ๐ } X ( ๐ | ๐ฅ๐ฆ ) X ( ๐ | ๐ฆ ) { ๐ฅ, ๐ฆ } { ๐ฆ } { ๐ฆ } { ๐ฅ ๐ , ๐ฆ } Figure 21: Geometric interpretation of a set-phenomenon renumberingthe terraced events, generated by the doublet of half-rare events X = { ๐ฅ, ๐ฆ } = { ๐ฅ, ๐ฆ } ( ๐ | ๐ฅ๐ฆ ) , by re๏ฌections 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 ( ๐ | ๐ฅ๐ฆ ) and its set-phenomena X ( ๐ | ๐ฅ ) = { ๐ฅ, ๐ฆ ๐ } , X ( ๐ | ๐ฆ ) = { ๐ฅ ๐ , ๐ฆ } and X ( ๐ |โ ) = { ๐ฅ ๐ , ๐ฆ ๐ } 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 ๐ โ X . The degenerate e.p.d. of the1st kind with ๐ ( ๐// X ) = 1 corresponds to this ver-tex. And associate the vertex ๐ ๐๐๐๐ with the hy-percube [0 , ๐๐ก๐๐๐๐ ๐๐๐๐๐ , numbered by the ๐ ๐๐๐๐ -set:, e.p.d. , , , of the 1st kind, , . ๐ โ X X - S X , e.p.d. of the 1st kind ๐ ( ๐// X ) = 1 . And associatewith it the vertex of X -hypercube [0 , โ X , number-ing by the following X -set: { ฮฅ ๐// X ( ๐ฅ ) , ๐ฅ โ X } (11.43)where ฮฅ ๐// X ( ๐ฅ ) = {๏ธ , ๐ฅ โ ๐, , , (11.44)are values of the indicator of subset ๐ โ X onevents ๐ฅ โ X . De๏ฌne a prohection pr : S X โ [0 , โ X of the X -simplex S X on X -hypercube [0 , โ X by thefollowing formula: pr ( { ๐ ( ๐// X ) , ๐ โ X } ) = { ๐ ๐ฅ , ๐ฅ โ X } (11.45) where ๐ ๐ฅ = โ๏ธ ๐ โ X ๐ ( ๐// X )ฮฅ ๐// X ( ๐ฅ ) == โ๏ธ ๐ฅ โ ๐ โ X ๐ ( ๐// X ) (11.46)is a convex combination of hypercube vertices,which, as known [1], is intepreted as the probabil-ity of event ๐ฅ โ 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 ๐ โ X corresponds to thedegenerate e.p.d. of the 1st kind with probabilities ๐ ( ๐// X ) = {๏ธ , ๐ = ๐ , , ๐ ฬธ = ๐ โ X . From (11.46) you obtain that ๐ ๐ฅ = ฮฅ ๐ // X ( ๐ฅ ) = {๏ธ , ๐ฅ โ ๐ , , ๐ฅ โ X โ ๐ . Therefore, by (11.43) { ๐ ๐ฅ , ๐ฅ โ X } = { ฮฅ ๐ // X ( ๐ฅ ) , ๐ฅ โ X } is a vertex of the X -hypercube.In particular, the โ -vertex of the X -simplex, i.e., thevertex numbered by the subset ๐ = โ is projectedinto the X -set { , ..., } , consisting of the zero prob-abilities of marginal events, in other words, pro-jected into the โ -vertex of the X -hypercube, i.e., tothe vertex located at the beginning coordinates: { , ..., } โผ ๐ ( ๐// X ) = {๏ธ , ๐ = โ , , โ ฬธ = ๐ โ X ; (11.47)and the X -vertex of the X -simplex, i.e., the vertexenumerated by the subset ๐ = 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: { , ..., } โผ ๐ ( ๐// X ) = {๏ธ , ๐ = X , , X ฬธ = ๐ โ 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.
18 T HE XIV FAMEMSโ2015 C
ONFERENCE ( ๐ โ -diagram We will ๏ฌgure out how a Venn ( ๐ โ -diagram ofan arbitrary ( ๐ โ -s.e. is constructed on the ba-sis of the projection (11.45), in which the role of thespace of universal elementary events โฆ is playedby the unit ( ๐ โ โdimensional hypercube . Such aVenn ( ๐ โ -diagram puts terraced hypercubes gen-erated by dividing a unit hypercube in half orthog-onal to each of the ๐ axes into a one-to-one corre-spondence with the terraced events generated bythe given ( ๐ โ -s.e.Take ๏ฌrst ( ๐ โ -s.h-r.e. X and represent its Venn ( ๐ โ -diagram On which โฆ is represented by a unit ( ๐ โ -dimensional hypercube that serves as or-dered image of the X -hypercube [0 , โ X = โจ๏ธ ๐ฅ โ X [0 , ๐ฅ , (11.49)broken by hyperplanes orthogonal to ๐ฅ -axis and in-tersecting them at points / into ๐ ๐ -terraced hy-percubes for ๐ โ X [0 , โ ter ( ๐// X ) = โจ๏ธ ๐ฅ โ ๐ [0 , / ๐ฅ โจ๏ธ ๐ฅ โ X โ ๐ (1 / , ๐ฅ , (11.50)where each marginal half-rare event ๐ฅ โ X is rep-resented as a ๐ฅ -half of X -hypercube containing theorigin : [0 , / ๐ฅ โ [0 , โ ( X โ{ ๐ฅ } ) , (11.51)its complement ๐ฅ ๐ = โฆ โ ๐ฅ is represented in theform of another ๐ฅ -half of X -hypercube that does notcontain the origin : (1 / , ๐ฅ โ [0 , โ ( X โ{ ๐ฅ } ) , (11.52)and the ๐ -terraced event ter ( ๐// X ) โ as a ๐ -terraced hypercube (11.50): ter ( ๐// X ) โผ [0 , โ ter ( ๐// X ) . (11.53)The formula (11.53) once again points to a one-to-one correspondence between the ๐ -space of ter-raced hypercubes (11.50) from the Venn ( ๐ โ -diagram of ( ๐ โ -s.h-r.e. X and ๐ -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 de๏ฌnedby 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 ๐ is discussed in [ ? ]. Note 22 (correspondence between the numbering of ter-raced hypercubes and vertices of X -simplex of e.p.d.โsof the 1st kind of ( ๐ โ -s.h-r.e. X on its Venn ( ๐ โ -diagram). On the Venn ( ๐ โ -diagram of ( ๐ โ -setof half-rare events X , every ๐ ๐ -terraced hyper-cube [0 , โ ter ( ๐ ๐ // X ) contains the ๐ -vertex of X -hypercube, into which corresponding ๐ -vertexof X -simplex S X of e.p.d.โs of the 1st kind of ( ๐ โ -s.h-r.e. X for ๐ โ X is projected: { , ๐ฅ โ ๐ } + { , ๐ฅ โ X โ ๐ } โ [0 , โ ter ( ๐ ๐ // X ) , (11.54)in particular, for โ -vertex and X -vertex we have: { , ๐ฅ โ X } = { , ..., } โ [0 , โ ter ( X // X ) , { , ๐ฅ โ X } = { , ..., } โ [0 , โ ter ( โ // X ) . (11.55), ( ๐ โ - - ..ยด. ( ๐ โ - X events [6].It is not di๏ฌcult to guess that the Venn ( ๐ โ -diagram of an arbitrary set-phenomenon of s.h-r.e.differs from the Venn ( ๐ โ -diagram of X itselfonly by renumbering terraced events using formu-las from [6]. Letโs do Note 23 (Venn ( ๐ โ -diagram of set-phenomena of a setof half-rare events). On the Venn ( ๐ โ -diagramof the ๐ -phenomenon X ( ๐ | ๐ ) of ( ๐ โ -set of half-rare events X , every ๐ โ ๐ -terraced hyper-cube [0 , โ ter ( ๐ ฮ ๐// X ) contains the ๐ -vertex of X -hypercube, into which the corresponding ๐ -vertexof X -simplex S X of e.p.d. of the 1st kind of ( ๐ โ -s.h-r.e. X for ๐ โ X and ๐ โ X is projected: { , ๐ฅ โ ๐ } + { , ๐ฅ โ X โ ๐ } โ [0 , โ ter ( ๐ ฮ ๐// X ) , (11.56)in particular, for the โ -phenomenon X ( ๐ |โ ) = X ( ๐ ) and for โ -vertex and X -vertex we have: { , ๐ฅ โ X } = { , ..., } โ [0 , โ ter ( โ // X ( ๐ ) ) , { , ๐ฅ โ X } = { , ..., } โ [0 , โ ter ( X ( ๐ ) // X ( ๐ ) ) . (11.57) X -hypercube Definition 11 ( set-phenomenon spectrum of func-tions on the X -hypercube ). With each function ๐ โ ฮจ X , de๏ฌned on the X -hypercube, the ๐ func-tions are connected. These functions are de๏ฌnedon the X -hypercube by formulas: ๐ ๐ ( ห ๐ค ) = ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ for ๐ โ X . The family of the all such functions { ๐ ๐ : ๐ โ X } is called the set-phenomenon X -spectrum of thefunction ๐ . OROBYEV { ๐ฆ } { ๐ฅ, ๐ฆ } ๐ ๐ฅ (cid:62) โ ๐ ๐ฆ โ ๐ ๐ฅ (cid:62) โ ๐ ๐ฆ โ ๐ ๐ฆ > ๐ ๐ฅ โ ๐ ๐ฆ > โ ๐ ๐ฅ ๐ ๐ฆ > ๐ ๐ฅ ๐ ๐ฆ > โ ๐ ๐ฅ ๐ ๐ฅ (cid:62) ๐ ๐ฆ โ ๐ ๐ฅ (cid:62) ๐ ๐ฆ โ { ๐ฅ } ter ( ๐ฅ//๐ฅ๐ฆ ) ter ( โ //๐ฅ๐ฆ ) ter ( ๐ฆ//๐ฆ๐ฅ ) ter ( โ //๐ฆ๐ฅ ) ter ( ๐ฆ๐ฅ//๐ฆ๐ฅ ) ter ( ๐ฅ//๐ฆ๐ฅ ) ter ( ๐ฅ๐ฆ//๐ฅ๐ฆ ) ter ( ๐ฆ//๐ฅ๐ฆ ) Figure 22: The projection of a simplex (tetrahedron) of doublets events S { ๐ฅ,๐ฆ } on a unit { ๐ฅ, ๐ฆ } -square [0 , โ{ ๐ฅ,๐ฆ } of its marginal probabilities ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ } . The ๐ - of this simplex are projected in corresponding ๐ -vertices of { ๐ฅ, ๐ฆ } -square, ๐ โ { ๐ฅ, ๐ฆ } , and the all of e.p.d.โs of dou-blets of events with given { ๐ฅ, ๐ฆ } -set of probabilities of marginal events ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ } are projected in each point ห ๐ โ [0 , โ{ ๐ฅ,๐ฆ } . In the leftdown quadrant (aqua) e.p.d.โs of the all of doublets of half-rare events oftwo kind are projected: / (cid:62) ๐ ๐ฅ (cid:62) ๐ ๐ฆ (unshaded) and ๐ ๐ฅ < ๐ ๐ฆ (cid:54) / (shaded triangle); in the remaining 3 quadrants e.p.d.โs of โ -phenomena, { ๐ฆ } -phenomena and { ๐ฅ } -phenomena of doublets of half-rare events areprojected. The half-rare doublets of the second kind: ๐ ๐ฅ < ๐ ๐ฆ (cid:54) / ,are projected in the shaded triangle of left down quadrant, and its set-phenomena โ in shaded triangles of corresponding quadrants. The ter-raced events, generated by doublets of half-rare events { ๐ฅ, ๐ฆ } , are markedby the white formulas. Letโs de๏ฌne for each ๐ โ X the terraced ๐ -hypercube ter โ ( ๐// X ) = โจ๏ธ ๐ฅ โ ๐ [๏ธ , ]๏ธ โจ๏ธ ๐ฅ โ X โ ๐ [๏ธ , )๏ธ , from which the X -hypercube is composed: [0 , โ X = โ๏ธ ๐ โ X ter โ ( ๐// X ) . Lemma 6 ( on a set-phenomenon X -spectrum of nor-malized function ). In order that the family offunctions { ๐ ๐ : ๐ โ X } ฮจ X is a set-phenomenon X -spectrum of some function normalized on the X -hypercube, it is necessary and su๏ฌcient that โ๏ธ ๐ โ X ๐ ๐ ( ห ๐ค ) = 1 (11.58)for all ห ๐ค โ [0 , โ X . Proof.
1) If the family { ๐ ๐ : ๐ โ X } is aset-phenomenon X -spectrum of some normalizedfunction, then by De๏ฌnition 11 the equality (11.58)is satis๏ฌed. 2) Let now the equality (11.58) is satis-๏ฌed. Construct the function ๐ on the X -hypercube { ๐ฆ } { ๐ฅ, ๐ฆ } ๐ ๐ฅ (cid:62) โ ๐ ๐ฆ โ ๐ ๐ฅ (cid:62) โ ๐ ๐ฆ โ ๐ ๐ฆ > ๐ ๐ฅ โ ๐ ๐ฆ > โ ๐ ๐ฅ ๐ ๐ฆ > ๐ ๐ฅ ๐ ๐ฆ > โ ๐ ๐ฅ ๐ ๐ฅ (cid:62) ๐ ๐ฆ โ ๐ ๐ฅ (cid:62) ๐ ๐ฆ โ { ๐ฅ } { ๐ฅ } โ { ๐ฆ } โ { ๐ฆ, ๐ฅ } { ๐ฅ }{ ๐ฅ, ๐ฆ } { ๐ฆ } Figure 23: The projection of a simplex (tetrahedron) of doublets events S { ๐ฅ,๐ฆ } on a unit { ๐ฅ, ๐ฆ } -square [0 , โ{ ๐ฅ,๐ฆ } of its marginal probabilities ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ } . The ๐ - of this simplex are projected in corresponding ๐ -vertices of { ๐ฅ, ๐ฆ } -square, ๐ โ { ๐ฅ, ๐ฆ } , and the all of e.p.d.โs of dou-blets of events with given { ๐ฅ, ๐ฆ } -set of probabilities of marginal events ห ๐ = { ๐ ๐ฅ , ๐ ๐ฆ } are projected in each point ห ๐ โ [0 , โ{ ๐ฅ,๐ฆ } . In the leftdown quadrant (aqua) e.p.d.โs of the all of doublets of half-rare events oftwo kind are projected: / (cid:62) ๐ ๐ฅ (cid:62) ๐ ๐ฆ (unshaded) and ๐ ๐ฅ < ๐ ๐ฆ (cid:54) / (shaded triangle); in the remaining 3 quadrants e.p.d.โs of โ -phenomena, { ๐ฆ } -phenomena and { ๐ฅ } -phenomena of doublets of half-rare events areprojected. The half-rare doublets of the second kind: ๐ ๐ฅ < ๐ ๐ฆ (cid:54) / ,are projected in the shaded triangle of left down quadrant, and its set-phenomena โ in shaded triangles of corresponding quadrants. The ter-raced events, generated by doublets of half-rare events { ๐ฅ, ๐ฆ } , are markedby the white formulas. by the following way ๐ ( ห ๐ค ) = โงโชโชโชโชโจโชโชโชโชโฉ ๐ โ (๏ธ ห ๐ค ( ๐ |โ ) )๏ธ , ห ๐ค โ [0 , / โ X ,. . . ,๐ ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ , ห ๐ค โ ter โ ( ๐// X ) ,. . . ,๐ X (๏ธ ห ๐ค ( ๐ | X ) )๏ธ , ห ๐ค โ [1 / , โ X and show that the ๐ is normalized on the X -hypercube. Indeed, noting that for an arbitrary ๐ โ X the equality ๐ ( ห ๐ค ) = ๐ ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ is equiva-lent to the equality ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = ๐ ๐ ( ห ๐ค ) , we obtainthe required: โ๏ธ ๐ โ X ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = โ๏ธ ๐ โ X ๐ ๐ ( ห ๐ค ) = 1 . Lemma 7 ( on a set-phenomenon X -spectrum of the1-function ). In order that the family of functions { ๐ ๐ : ๐ โ X } ฮจ X is a set-pehomenon X -spectrum ofsome 1-function on the X -hypercube, it is necessaryand su๏ฌcient that for each ๐ฅ โ X โ๏ธ ๐ฅ โ ๐ โ X ๐ ๐ ( ห ๐ค ) = ๐ค ๐ฅ (11.59)
20 T HE XIV FAMEMSโ2015 C
ONFERENCE for all ห ๐ค โ [0 , โ 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 ๐ฅ โ ๐ โ X ๐ค ๐ฅ : โ๏ธ ๐ฅ โ ๐ โ X ๐ ๐ ( ห ๐ค ) = ๐ค ๐ฅ for each ห ๐ค โ [0 , โ X by De๏ฌnition 4. 2) Let nowthe equalities (1.4) are satis๏ฌed. Letโs construct thefunction ๐ on X -hypercube by the following way ๐ ( ห ๐ค ) = โงโชโชโชโชโจโชโชโชโชโฉ ๐ โ (๏ธ ห ๐ค ( ๐ |โ ) )๏ธ , ห ๐ค โ [0 , / โ X ,. . . ,๐ ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ , ห ๐ค โ ter โ ( ๐// X ) ,. . . ,๐ X (๏ธ ห ๐ค ( ๐ | X ) )๏ธ , ห ๐ค โ [1 / , โ X and show that ๐ is a 1-function on the X -hypercube.Indeed, noting that for an arbitrary ๐ โ X theequality ๐ ( ห ๐ค ) = ๐ ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ is equivalent to theequality ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = ๐ ๐ ( ห ๐ค ) , we obtain the re-quired: โ๏ธ ๐ฅ โ ๐ โ X ๐ (๏ธ ห ๐ค ( ๐ | ๐// X ) )๏ธ = โ๏ธ ๐ฅ โ ๐ โ X ๐ ๐ ( ห ๐ค ) = ๐ค ๐ฅ . { , } { , } ๐ฅ๐ฆ ๐ฅ๐ฆ
10 00 ๐ฆ๐ฅ ๐ฆ๐ฅ
01 00 ๐ฆ๐ฅ ๐ฆ๐ฅ
11 10 ๐ฅ๐ฆ ๐ฅ๐ฆ
11 01 { ๐ ๐ฅ , ๐ ๐ฆ } = { , } { , } { ๐ฅ } โ { ๐ฅ,๐ฆ } { ๐ฆ } 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 ห ๐ -ordering condition of the set of { , , } { , , } ๐ฅ๐ฆ๐ง ๐ฅ๐ฆ๐ง
100 000 ๐ฅ๐ง๐ฆ ๐ฅ๐ง๐ฆ
100 000 ๐ฆ๐ฅ๐ง ๐ฆ๐ฅ๐ง
010 000 ๐ง๐ฅ๐ฆ ๐ง๐ฅ๐ฆ
010 000 ๐ฆ๐ง๐ฅ ๐ฆ๐ง๐ฅ
001 000 ๐ง๐ฆ๐ฅ ๐ง๐ฆ๐ฅ
001 000 ๐ง๐ฆ๐ฅ ๐ง๐ฆ๐ฅ
011 010 ๐ฆ๐ง๐ฅ ๐ฆ๐ง๐ฅ
101 100 ๐ง๐ฅ๐ฆ ๐ง๐ฅ๐ฆ
011 001 ๐ฆ๐ฅ๐ง ๐ฆ๐ฅ๐ง
110 100 ๐ฅ๐ง๐ฆ ๐ฅ๐ง๐ฆ
101 001 ๐ฅ๐ฆ๐ง ๐ฅ๐ฆ๐ง
110 010 { ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง } = { , , } { , , } { ๐ฅ } โ { ๐ฅ,๐ฆ } { ๐ฆ } { , , } { , , } ๐ฅ๐ฆ๐ง ๐ฅ๐ฆ๐ง
101 001 ๐ฅ๐ง๐ฆ ๐ฅ๐ง๐ฆ
110 010 ๐ฆ๐ฅ๐ง ๐ฆ๐ฅ๐ง
011 001 ๐ง๐ฅ๐ฆ ๐ง๐ฅ๐ฆ
110 100 ๐ฆ๐ง๐ฅ ๐ฆ๐ง๐ฅ
011 010 ๐ง๐ฆ๐ฅ ๐ง๐ฆ๐ฅ
101 100 ๐ง๐ฆ๐ฅ ๐ง๐ฆ๐ฅ
111 110 ๐ฆ๐ง๐ฅ ๐ฆ๐ง๐ฅ
111 110 ๐ง๐ฅ๐ฆ ๐ง๐ฅ๐ฆ
111 101 ๐ฆ๐ฅ๐ง ๐ฆ๐ฅ๐ง
111 101 ๐ฅ๐ง๐ฆ ๐ฅ๐ง๐ฆ
111 011 ๐ฅ๐ฆ๐ง ๐ฅ๐ฆ๐ง
111 011 { ๐ ๐ฅ , ๐ ๐ฆ , ๐ ๐ง } = { , , } { , , } { ๐ฅ,๐ง } { ๐ง }{ ๐ฅ,๐ฆ,๐ง } { ๐ฆ,๐ง } 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 โ at thebottom. In the Venn diagram of half-rare events: ๐ฅ is the left, ๐ฆ is theright, and ๐ง is the lower half of the cube. events is speci๏ฌed, 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 su๏ฌcientto 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 ๐ -Kopula in R ๐ is su๏ฌcient to de๏ฌne onthe /๐ ! part of the unit ๐ -hypercube so that thisrepresentation becomes de๏ฌnite 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 ๐ -Kopula defines ๐ classical copulas of ๐ marginal uniform distributions on [0 , ).
13 On the inevitable development oflanguage
This ๏ฌrst 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-๏ฌected 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 โ 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 not๏ฌnd any worthy re๏ฌection. 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 โ 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 rยดepartition ยดa ๐ dimensions et leursmarges. Publ. Inst. Statist. Univ. Paris , 8:229โ231, 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 10โ28, 2003. [78][5] C. Alsina, M.J. Frank, and B. Schweizer.
Assocative func-tions: Triangular Norms and Copulas . World Scienti๏ฌc 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 52โ59, ISBN 978โ5โ9903358โ5โ1, , 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.
Notes of Krasnoyarsk State University, Phys. Math. Series ,1:152โ159, 2006. [79, 91][8] O. Yu. Vorobyev. On the foundations of the eventologicalmethod. In.
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 112โ121, ISBN 978โ5โ9903358โ4โ4, 2014. [81, 111][9] O. Yu. Vorobyev. Frยดechet boundary eventological distri-butions and its applications. In.
Proc. of the XV Intern.EM Conf. on Eventological Mathematics. Krasnoyarsk, KSU(Oleg Vorobyev ed.) , 1:57โ69, 2010. [117][10] O. Yu. Vorobyev. Eventologial generalization of Frยดechetbounds 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 122โ130, ISBN 978โ5โ9903358โ4โ4, 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 78โ125, ISBN 978โ5โ9903358โ5โ1, , 2015. [121]
22 T HE XIV FAMEMSโ2015 C
ONFERENCE [12] O. Yu. Vorobyev. Theory of dual co โผ event means. In. Proc.of the XV Intern. FAMEMS Conf. on Financial and Actuar-ial Mathematics and Eventology of Multivariate Statistics &the Workshop on Hilbertโs Sixth Problem; Krasnoyarsk, SFU(Oleg Vorobyev ed.) , pages 44โ93, ISBN 978โ5โ9903358โ6โ8, , 2016. [121][13] O. Yu. Vorobyev. Postulating the theory of experience andof chance as a theory of co โผ events (co โผ beings). In. Proc.of the XV Intern. FAMEMS Conf. on Financial and Actuar-ial Mathematics and Eventology of Multivariate Statistics &the Workshop on Hilbertโs Sixth Problem; Krasnoyarsk, SFU(Oleg Vorobyev ed.) , pages 25โ43, ISBN 978โ5โ9903358โ6โ8, , 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 โผ events. In. Proc. of the XV Intern. FAMEMS Conf.on Financial and Actuarial Mathematics and Eventology ofMultivariate Statistics & the Workshop on Hilbertโs SixthProblem; Krasnoyarsk, SFU (Oleg Vorobyev ed.) , pages 9โ24,ISBN 978โ5โ9903358โ6โ8,