Maximal Non-Exchangeability in Dimension d
aa r X i v : . [ m a t h . S T ] N ov MAXIMAL NON-EXCHANGEABILITY IN DIMENSION D MICHAEL HARDER AND ULRICH STADTM ¨ULLER
Abstract.
We give the maximal distance between a copula and itself when the argument ispermuted for arbitrary dimension, generalizing a result for dimension two by Nelsen (2007);Klement and Mesiar (2006). Furthermore, we establish a subset of [0 , d in which this boundmight be attained. For each point in this subset we present a copula and a permutation, forwhich the distance in this point is maximal. In the process, we see that this subset depends onthe dimension being even or odd. Introduction
Studying the dependence structure in the distribution function H of a d -dimensional continuousrandom vector X the so called copula is crucial. This is the distribution C of the random vector U with components U i = F i ( X i ) where F i is the one-dimensional marginal distribution of X i . For details, see Sklar’s Theorem in Sklar (1959).Of interest are in particular parametric classes of such copulas. The usual examples, how-ever, have the disadvantage that they share some symmetry properties. Quite popular areArchimedean copulas which have the form C ( u , ..., u d ) = ϕ ( ϕ − ( u ) + . . . , ϕ − ( u d )) , with a generating function ϕ ( s ) being most often the Laplace transform of a distribution on(0 , ∞ ). If these generating functions contain some parameter θ we are given a parametric cop-ula model. However, a random vector U having this copula as a distribution has exchangeablecomponents. But it is not clear whether data which have to be investigated follow an exchange-able copula. On the way to look for tests on exchangeability one comes across the question:what is the maximal distance between a copula and a version of it where the arguments arepermuted. This paper is devoted to this question.In the following, let d ∈ N \ { } denote the dimension. Definition 1.1.
A random vector X := ( X , . . . , X d ) ⊤ is called exchangeable , if its law coincideswith the law of the random vector X π := ( X π (1) , . . . , X π ( d ) ) ⊤ , where π ∈ S d is a permutation of { , . . . , d } .Let H be the cdf of X and H π the cdf of X π . Then it is straightforward to see, that if X isexchangeable, then all marginal cdfs must be identical. Definition 1.2.
A mapping F : R d R is called exchangeable , if F ( x , . . . , x d ) = F ( x π (1) , . . . , x π ( d ) )holds for all ( x , . . . , x d ) ⊤ ∈ R d and all permutations π ∈ S d .Note, that instead of exchangeable the notion symmetric is used as well (e. g. for aggregationfunctions by Grabisch et al. (2009)), which however is not used in a uniquely defined way (e. g.Nelsen (1993) defines four different kinds of symmetry of a distribution function). It may seemunusual to use the same word for a property of a random vector as well as for a property of amapping. But it is easy to verify that a random vector is exchangeable if and only if its cdf isexchangeable. From the famous theorem by Sklar (1959) it follows that a multivariate cumulativedistribution function is exchangeable if and only if its copula is exchangeable (provided that all Preprint submitted to Journal of Multivariate Analysis marginal cdfs are identical). In the following, we will address the exchangeability—or rather thelack of this property—of copulas.Now, being interested in statistical tests to decide whether some data come from an exchangeablecopula it is important to know how big the difference of a copula from itself with permutedcomponents can be. For exchangeable copulas this difference is zero. Here comes the first resultin this direction.Nelsen (2007) shows that for d = 2 and any copula C it holds that(1) | C ( u ) − C ( u π ) | ≤
13 for all u ∈ [0 , and all π ∈ S . The same result has been published independently by Klement and Mesiar (2006). For π = idobviously C ( u ) = C ( u π ), so for d = 2 there’s only one interesting permutation, namely π = τ (1 , u and u . The bound in (1) is the best possible, as Nelsen(2007) demonstrates by showing that C ( u , u ) := min (cid:26) u , u , (cid:16) u − (cid:17) + + (cid:16) u − (cid:17) + (cid:27) is a copula and for u := (cid:0) , (cid:1) ⊤ the bound in (1) is attained. As usual we denote by f + :=max { f, } . By defining ˜ C ( u , u ) := C ( u , u ) for any ( u , u ) ⊤ ∈ [0 , , we obviously get another copula˜ C . Therefore, (1) could be rewritten as(2) max u ∈ [0 , | C ( u ) − ˜ C ( u ) | ≤ .
5, as | C a ( u ) − C b ( u ) | ≤ M ( u ) − W ( u ) ≤ M (cid:0) , (cid:1) − W (cid:0) , (cid:1) = 12shows, where M ( u , u ) := min { u , u } and W ( u , u ) := max { , u + u − } are the upperand lower Fr´echet-Hoeffding-bounds, respectively. Note that this bound is best possible sinceit is attained by the two copulas M and W .
Whereas the extension of the latter inequality toarbitrary dimension d is obvious this is not the case for the inequality (1). Hence, it is aim ofthe present paper is to extend inequality (1) to arbitrary dimension d and to investigate thecopulas and the set of points where this bound is attained.2. Main Result
Now, let’s state the main theorem of this paper, generalizing the inequality (1) to arbitrary di-mension d . Just like in Definition 1.1, given a vector u ∈ [0 , d , we write u π := ( u π (1) , . . . , u π ( d ) ) ⊤ for the vector whose components are permuted according to π ∈ S d . Theorem 1.
Let C be a d -copula. Then (3) max u ∈ [0 , d | C ( u ) − C ( u π ) | ≤ d − d + 1 holds true for any permutation π ∈ S d . The bound is best possible, i.e. for each dimension d thereexists a d -copula C , a permutation π ∈ S d and a vector u ∗ ∈ [0 , d , such that | C ( u ∗ ) − C ( u ∗ π ) | = d − d +1 . Remarks: i) The difference between two arbitrary copulas C and C of dimension d can be bounded for all u ∈ [0 , d as follows | C ( u ) − C ( u ) | ≤ M d ( u ) − W d ( u ) ≤ M d ( u ∗ ) − W d ( u ∗ ) = d − d AXIMAL NON-EXCHANGEABILITY IN DIMENSION D with the Fr´echet-Hoeffding-bounds M d ( u ) = min { u , . . . , u d } and W d ( u ) = max { P di =1 u i − d +1 , } , and u ∗ j = ( d − /d for all j = 1 , . . . , d . Although W d is no copula for d > , the bound d − d is best possible, since for every fixed u ∈ [0 , d there exists a copula C , such that C ( u ) = W d ( u ) (see e. g. Nelsen (2006) or for an exact form of such a copula with given diagonal section, seeJaworski (2009)).ii) If we assume u ∗ ≤ u ∗ , Nelsen (2007) shows that for d = 2 there is exactly one u ∗ = (cid:0) , (cid:1) ⊤ for which the maximum in (3) is attained. Under the condition, that u ∗ ≤ · · · ≤ u ∗ d , we getnonuniqueness or uniqueness of u ∗ depending on d being even or odd. For d = 2 n + 2 , n ∈ N there are infinitely many choices for such a u ∗ —yet within some lower dimensional manifold.In any case, for d > , a fixed u ∗ and a fixed copula C , such that the bound in (3) is achieved,there’s still more than one choice for the permutation π . This will be discussed in more detailin Section 4.iii) Based on our result we could define µ ( C ) := d + 1 d − π ∈ S d max u ∈ [0 , d | C ( u ) − C ( u π ) | as a measure of non-exchangeability for the copula C . Note, that the definition of measuresof non-exchangeability by Durante et al. (2010) is just for bivariate copulas and therefore notapplicable in this case. In the following corollary we see that Theorem 1 is not just a statement about exchangeability,but also has consequences for the possible choices of lower dimensional margins of a copula. Forexample, if d > d − C a and C b coincide on the point d − d − (1 , . . . , ⊤ with the Fr´echet-Hoeffding-bounds. Corollary 2.1.
Let d > , C be a d -copula and ≤ k < d − . Let C ( d − k ) ,a and C ( d − k ) ,b two ( d − k ) -dimensional margins of C . Then | C ( d − k ) ,a ( ˜ u ) − C ( d − k ) ,b ( ˜ u ) | ≤ d − d + 1 < d − k − d − k = M d − k ( u ∗ ) − W d − k ( u ∗ ) for all ˜ u ∈ [0 , d − k and u ∗ := d − k − d − k (1 , . . . , ⊤ ∈ [0 , d − k By M d − k we denote the upper ( d − k )-dimensional Fr´echet-Hoeffding-bound, and by W d − k a( d − k )-copula which coincides with the lower ( d − k )-dimensional Fr´echet-Hoeffding-bound in u ∗ . Note, that Corollary 2.1 is still correct for d = 3, but gives no information. Proof. As C ( d − k ) ,a and C ( d − k ) ,b are margins of C , for a fixed ˜ u ∈ [0 , d − k there exist u a , u b ∈ [0 , d with exactly k components equal to 1, such that C ( d − k ) ,a ( ˜ u ) = C ( u a ) and C ( d − k ) ,b ( ˜ u ) = C ( u b ).These two d -dimensional vectors u a and u b are the same, up to the order of their components.Therefore, there exists a permutation π ∈ S d such that u a = ( u b ) π and | C ( d − k ) ,a ( ˜ u ) − C ( d − k ) ,b ( ˜ u ) | = | C ( u a ) − C (cid:0) ( u a ) π (cid:1) | ≤ d − d + 1 .The other equations are straightforward to compute. (cid:3) Proof of the main result
Before proving Theorem 1 we first state some auxiliary results needed in the proof. By τ ij wedenote the transposition of i and j , i.e. the permutation interchanging components i and j andleaving the others unchanged. Lemma 3.1.
Let u ∈ [0 , d , let i, j ∈ { , . . . , d } , then | C ( u ) − C ( u τ ij ) | ≤ | u i − u j | holds for any d -copula C . MICHAEL HARDER AND ULRICH STADTM ¨ULLER
Proof.
Let C be a d -copula, u ∈ [0 , d and i, j ∈ { , . . . , d } . Now define v by v k := max { u k , u τ ij ( k ) } , k = 1 , . . . , d which implies v k = u k for k = i, j . Due to the monotonicity of C we get(4) C ( u ) ≤ C ( v ) , C ( u τ ij ) ≤ C ( v ). C being Lipschitz-continuous (see e. g. Nelsen (2006)) yields(5) | C ( v ) − C ( u ) | ≤ d X k =1 | v k − u k | = | v i − u i | + | v j − u j | where the last equation is due to the choice of v . As v i = v j = max { u i , u j } either | v i − u i | or | v j − u j | vanishes. Together with (4) we conclude C ( u ) ∈ (cid:2) C ( v ) − | u i − u j | , C ( v ) (cid:3) .By replacing u in (5) by u τ ij , it is easy to see, that C ( u τ ij ) is within the same interval, whichcompletes the proof. (cid:3) In the next lemma, we will show that the upper inequality in Theorem 1 holds. For the proofwe need the following example of special permutations.
Example 3.1.
Let u ∈ [0 , d and π ∈ S d . Note that in this example each transposition mightbe the identity mapping. Let τ d be the transposition, which exchanges d and π ( d ). Thus, τ d puts u d in the right place. Now let τ d − be the transposition, which puts u d − in u τ d in the right place.If ( d −
1) wasn’t concerned by τ d (i.e. τ d ( d −
1) = d − τ d − is the transposition whichexchanges ( d −
1) and π ( d −
1) (note that π ( d − = π ( d ), so u d remains untouched). Otherwise, τ d ( d ) = d − τ d − ( d −
1) = d − τ d − ( d ) = π ( d − u d and u d − in the right places, i.e. on the same positions in u π and u τ d − ◦ τ d . Likethis, we can go on, until τ finally puts u into its place. We needn’t worry about u , becausewhen u , . . . , u d are all on their places, then u has to be taken care of as well. In a nutshell, π can be replaced by the composition of at most d − π : (1 , , , (3 , , , π is by π = τ ◦ τ ◦ τ , where the transpositions τ j are characterized by τ = (34) τ = (14) and τ = id.In this case, as τ = id, even two transpositions suffice to generate π = (143). Lemma 3.2.
Let u ∈ [0 , d , let π ∈ S d , then (6) | C ( u ) − C ( u π ) | ≤ d − d + 1 holds for any d -copula C .Proof. Let C be a d -copula. W.l.o.g. let u ≤ . . . ≤ u d , otherwise we replace C in the proof by˜ C with ˜ C ( v ) := C ( v σ − ) for all v ∈ [0 , d . Here σ ∈ S d is the permutation which orders thecomponents of u by size, i.e. u σ = ( u (1) , . . . , u ( d ) ) ⊤ .If there exists at least one i ∈ { , . . . , d } with u i < d − d +1 the claim follows immediately by | C ( u ) − C ( u π ) | ≤ max (cid:8) C ( u ) , C ( u π ) (cid:9) ≤ M ( u ) ≤ u i < d − d + 1 .Hence we may assume now that d − d +1 ≤ u . In the following, we write ˜ u i := u i − d − d +1 , so wehave 0 ≤ ˜ u i ≤ d +1 . The permutation π is generated by at most ( d −
1) transpositions (as
AXIMAL NON-EXCHANGEABILITY IN DIMENSION D described in Example 3.1, see also Dummit and Foote (2009)), therefore, we are able to write π = τ ◦ . . . ◦ τ d − ◦ τ d . Next we use the triangular inequality to derive | C ( u ) − C ( u π ) | ≤≤ | C ( u ) − C ( u τ d ) | + | C ( u τ d ) − C ( u τ d − ◦ τ d ) | + . . . + | C ( u τ ◦ ... ◦ τ d ) − C ( u π ) |≤ d X i =2 ( u i − u ) ≤ d X i =2 ˜ u i (7)where the second inequality follows from Lemma 3.1.At the same time, we have | C ( u ) − C ( u π ) | ≤ M d ( u ) − W d ( u ) ≤ u − (cid:16) d X i =1 u i − ( d − (cid:17) = 2 d − d + 1 − d X i =2 ˜ u i (8)with the Fr´echet-Hoeffding-bounds M d and W d (see Nelsen (2006)). Therefore, we may concludethat | C ( u ) − C ( u π ) | ≤ min (cid:26) d X i =2 ˜ u i , d − d + 1 − d X i =2 ˜ u i (cid:27) ≤ d − d + 1which completes the proof. (cid:3) In the proof of Lemma 3.2 we need u ≤ . . . ≤ u d just for notational convenience. Therefore, itis straightforward to derive the following corollary: Corollary 3.1.
With the prerequisites of Lemma 3.2 | C ( u ) − C ( u π ) | ≤ min (cid:26) u , . . . , u d , d X i =1 ( u i − u (1) ) , ( d −
1) + u (1) − d X i =1 u i (cid:27) holds for any d -copula C (where u (1) := min { u , . . . , u d } ). By now, we established the upper inequality in Theorem 1. In order to prove that it cannotbe improved, we have to find a proper d -copula, for which the bound in (3) is attained in somepoint u ∈ [0 , d and for some permutation π ∈ S d . To this end let u ∗ ∈ [0 , d such that(9) u ∗ j := d − d +1 for 1 ≤ j ≤ d +12 dd +1 for j = d + 1 and d even1 otherwisefor j ∈ { , . . . , d } . In the following we consider the mapping C ∗ : [0 , d → R with(10) C ∗ ( u ) := d − X j =0 d − ^ k =0 (cid:16) u (( j + k ) mod d )+1 − d X i =1 ,i I ( j,k ) (1 − u ∗ i ) (cid:17) + where I ( j, k ) := (cid:8) (( j + l ) mod d ) + 1 : l = 0 , , . . . , k (cid:9) and V di =1 a i := min { a , . . . , a d } . A smallcalculation shows that in the case d = 2 this copula satisfies C ∗ ( u , u ) = min { u , u , ( u − / + + ( u − / + } as discussed above. Lemma 3.3.
Let C ∗ be the mapping defined in (10) and u ∗ ∈ [0 , d as in (9) . Let π ∈ S d theorder reversing permutation, i.e. π ( k ) := d − k + 1 , then C ∗ ( u ∗ ) = 0 and C ∗ ( u ∗ π ) = d − d +1 .Proof. First, note that P di =1 u ∗ i = d − u ∗ and therefore, P di =1 (1 − u ∗ i ) = 1.Now for the first claim: let j ∈ { , . . . , d − } and k := 0. Thus I ( j, k ) = I ( j,
0) = { j + 1 } andbecause of d X i =1 ,i I ( j, (1 − u ∗ i ) = d X i =1 (1 − u ∗ i ) − (1 − u ∗ j +1 ) = u ∗ j +1 MICHAEL HARDER AND ULRICH STADTM ¨ULLER we get (cid:16) u ∗ (( j + k ) mod d )+1 − d X i =1 ,i I ( j,k ) (1 − u ∗ i ) (cid:17) + = ( u ∗ j +1 − u ∗ j +1 ) + = 0whenever k = 0. As this holds for each j we have C ∗ ( u ∗ ) = 0.In order to prove the second claim, note that C ∗ ( u ∗ ) = P d − j =0 V d − k =0 (cid:0) m j,k (cid:1) + , with m j,k := u ∗ ([( d − − ( j + k )] mod d )+1 − d X i =1 ,i I ( j,k ) (1 − u ∗ i )because d − (cid:0) (( j + k ) mod d ) + 1 (cid:1) + 1 = (cid:0) ( d − − ( j + k ) (cid:1) mod d + 1. Now let j ∈ { , . . . , d − } and 0 ≤ k ≤ d −
2. We want to show that m j,k is nondecreasing in k , i.e. m j,k ≤ m j,k +1 . Thisis the case if and only if α j,k := u ∗ ([( d − − ( j + k )] mod d )+1 − u ∗ ([( d − − ( j + k +1)] mod d )+1 ≤ − u ∗ (( j + k +1) mod d )+1 =: β j,k (11)holds. Obviously the left hand side of (11) is the difference between consecutive components of u ∗ , so α j,k = 0 for most choices of k . The cases where α j,k = 0 depend on d being odd or even.If d is even, α j,k = 0 if:(1) ([( d − − ( j + k )] mod d ) + 1 = 1. Then α j,k = u ∗ − u ∗ d < ≤ β j,k .(2) ([( d − − ( j + k )] mod d )+1 = d +2. In this case (( j + k +1) mod d )+1 = d − ( d +1)+1 = d .Therefore, α j,k = u ∗ d +2 − u ∗ d +1 = 1 d + 1 < d + 1 = 1 − u ∗ d = β j,k .(3) ([( d − − ( j + k )] mod d )+1 = d +1. In this case (( j + k +1) mod d )+1 = d − d +1 = d +1.Therefore, α j,k = u ∗ d +1 − u ∗ d = 1 d + 1 = 1 − u ∗ d +1 = β j,k .If d is odd, α j,k = 0 if:(1) see 1. where d is even.(2) ([( d − − ( j + k )] mod d )+1 = d +12 +1. In this case (( j + k +1) mod d )+1 = d − d +12 +1 = d +12 . Therefore, α j,k = u ∗ d +32 − u ∗ d +12 = 2 d + 1 = 1 − − u ∗ d +12 = β j,k .So we have α j,k ≤ β j,k and thus m j,k ≤ m j,k +1 for all choices of j and k . This means theminimum in (10) is always achieved for k = 0 which gives us C ∗ ( u ∗ ) = d − X j =0 ( m j, ) + = d − X j =0 ( u ∗ d − j − u ∗ j +1 ) + = d − d + 1as for j > d the term ( u ∗ d − j − u ∗ j +1 ) + is 0 by the construction of u ∗ . (cid:3) Now we are finally set to prove Theorem 1.
Proof of Theorem 1.
Let π ∈ S d and C be a d -copula. Then by Lemma 3.2 we get (3). InLemma 3.3 we show that there exists a point u ∗ ∈ [0 , d and a mapping C ∗ : [0 , d → R suchthat | C ∗ ( u ∗ ) − C ∗ ( u ∗ π ) | = d − d + 1 .So, all we need to do in order to prove Theorem 1 is to show that C ∗ is indeed a copula. Thisis the case, as it can be constructed as a shuffle of min. In two dimensions Mikusi´nski et al.(1992) show that by slicing the unit square vertically (including the mass of the upper Fr´echet-Hoeffding-bound on the main diagonal) and rearranging it, i.e. shuffling the strips, the resultingmass distribution will yield a proper copula. Mikusi´nski and Taylor (2010, section 6) state AXIMAL NON-EXCHANGEABILITY IN DIMENSION D that this also works for d > , d (with the mass on { u ∈ [0 , d | u = . . . = u d } ). [0 , d is separated along hyperplanes of the form { u k = λ k } . The separate partsare then rearranged. The resulting shuffle of the original mass distribution corresponds toa proper copula. C ∗ can be obtained this way, by using hyperplanes with λ k := P ki =1 (1 − u ∗ i ). Durante and Fern´andez-S´anchez (2010) generalize this concept by applying it to arbitrarycopulas. By Remark 2.1. therein, and following their notation, we get a copula ˜ C indicated by (cid:10) ( J k ) dk =1 , ( C i ) di =1 (cid:11) where C i ( u ) := M d ( u ) for i = 1 , . . . , d , and J k = ( J kj ) dj =1 with(12) J kj := (cid:2)P di =1 ,i = j,...,k (1 − u ∗ i ) , P di =1 ,i = j +1 ,...,k (1 − u ∗ i ) (cid:3) if j < k , (cid:2)P di =1 ,i = k (1 − u ∗ i ) , (cid:3) if j = k , (cid:2)P j − i = k +1 (1 − u ∗ i ) , P ji = k +1 (1 − u ∗ i ) (cid:3) if j > k ,for k = 1 , . . . , d . In Proposition 2.2. Durante and Fern´andez-S´anchez (2010) give an explicitexpression of ˜ C , namely(13) ˜ C ( u ) = d X j =1 λ ( J j ) M d (cid:18) ( u − a j ) + λ ( J j ) , . . . , ( u d − a dj ) + λ ( J j ) (cid:19) where a kj is the left limit of the interval J kj . Showing that ˜ C ( u ) = C ∗ ( u ) is just notationallydemanding. The sums in (10) and in (12) look similar, but in (10) we circumvent the distinctionof cases by using modular arithmetic. Note that in (13), we write ( u i − a ij ) + instead of u i − a ij in Proposition 2.2. in Durante and Fern´andez-S´anchez (2010). But from their proof it is clearthat a summand is 0 whenever u i < a ij for at least one i ∈ { , . . . , d } . (cid:3) Additional Results
As mentioned in Section 2, if we assume u ∗ ≤ u ∗ , Nelsen (2007) shows that for d = 2 there isexactly one u ∗ (namely u ∗ = (cid:0) , (cid:1) ⊤ ) for which the maximum in (3) is attained. For d > u ∗ , where equality in (3) holds, is unique if and only if d is odd (assumed u ∗ i ≤ u ∗ j for i ≤ j ). If d = 2 n + 2 ( n ∈ N ), then there is a ( d − M ⊂ [0 , d , such thatfor all u ∗ ∈ M , there exist a copula C and a permutation π ∈ S d with | C ( u ∗ ) − C ( u ∗ π ) | = d − d +1 .This is shown in Lemma 4.2. For the proof we are going to improve the bound in (7) which wasderived in the proof of Lemma 3.2. Lemma 4.1.
Let d ≥ and u ∈ [0 , d with u i ≤ u j for i ≤ j . Then (14) | C ( u ) − C ( u π ) | ≤ d X i = ⌈ d ⌉ +1 ( u i − u ) holds for any copula C and any permutation π ∈ S d , where ⌈ a ⌉ denotes the smallest integer n ≥ a . Before the proof of Lemma 4.1 for an arbitrary π , we will give the proof for a special case in thefollowing example. Example 4.1.
Let d ≥ u as in Lemma 4.1 and π ∈ S d such that π ( i ) = i for exactlythree i ∈ { , . . . , d } . This means, there are exactly three components u i , u i , u i in u , which arepermuted in u π . W. l. o. g. we may assume i < i < i . As π can’t be a transposition (otherwise,there is one k with π ( i k ) = i k ), either π is a left-shift or a right shift, i. e. π = π l := ( i i i )or π = π r := ( i i i ) (as there are no other derangements in S ). Now let τ := ( i i ) and τ := ( i i ), then π l and π r are generated by those two transpositions in the following way: π l = τ ◦ τ , π r = τ ◦ τ .So we have | C ( u ) − C ( u π l ) | ≤ | C ( u ) − C ( u τ ) | + | C ( u τ ) − C ( u π l ) || C ( u ) − C ( u π r ) | ≤ | C ( u ) − C ( u τ ) | + | C ( u τ ) − C ( u π l ) | MICHAEL HARDER AND ULRICH STADTM ¨ULLER and applying Lemma 3.1 yields | C ( u ) − C ( u π l ) | ≤ | u i − u i | + | u i − u i | = | u i − u i | ≤ | u d − u || C ( u ) − C ( u π r ) | ≤ | u i − u i | + | u i − u i | = | u i − u i | ≤ | u d − u | .Note that the last equation holds, as u ≤ u i ≤ u i ≤ u i ≤ u d by the prerequisites. Now, inthis special case, (14) follows immediately, as either π = π l or π = π r .For more information on generating permutations by transpositions, see e. g. Dummit and Foote(2009). We will make use of Example 4.1 in the following proof of Lemma 4.1. Proof.
Let d ≥ u ∈ [0 , d with u i ≤ u j for i ≤ j and π ∈ S d . We will need p ∈ N , defined by p := |{ ≤ i ≤ d : π ( i ) = i }| i. e. p is the number of elements of { , . . . , d } , which are no fixed points of π . Note, that for p = 0, there is nothing to show and p = 1 is impossible. Therefore, we may assume p ≥ p indices 1 ≤ i < . . . < i p ≤ d with π ( i k ) = i k for k ∈ { , . . . , p } . We will proof Lemma4.1 by establishing the similar claim(15) | C ( u ) − C ( u π ) | ≤ p X k = ⌈ p ⌉ +1 ( u i k − u i ).Then (14) follows immediately, as p X k = ⌈ p ⌉ +1 ( u i k − u i ) ≤ d X i = ⌈ d ⌉ +1 ( u i − u )holds true for all p and the corresponding index sets.The proof of (15) will be an induction on p . For p = 2 equation (15) holds true due to Lemma3.1. Now assume (15) holds for p − p ≥ y in i y := π ( i p ). In any case y = p as i p is by definition no fixed point of π . Case . y ∈ (cid:8)(cid:6) p (cid:7) + 1 , . . . , p − (cid:9) : Just like in Example 3.1, we can see π as a composition of atmost p − i k is put in its place, starting with i p . Therefore, wehave π = σ ◦ τ p , where τ p := (cid:0) i p π ( i p ) (cid:1) and σ is the permutation which is generated by all theremaining transpositions. As τ p ( i p ) = π ( i p ) by definition, i p is a fixed point of σ , so σ permutesjust p − | C ( u ) − C ( u π ) | ≤ | C ( u ) − C ( u σ ) | + | C ( u σ ) − C ( u π ) |≤ p − X k = ⌈ p − ⌉ +1 ( u i k − u i ) + ( u i p − u i y ) ≤ p X k = ⌈ p ⌉ +1 ( u i k − u i )by the induction hypothesis and Lemma 3.1 as l p − m + 1 ≤ y ≤ p − Case . p = 2 n + 1 and y = (cid:6) p (cid:7) = n + 1: Analogous to Case 1, as l p − m + 1 = n + 1. Case . p = 2 n and y = (cid:6) p (cid:7) = n : Now let i x := π − ( i p ) ( x = p as i p is not a fixed point of π ). Case . x > y : Similar to Case 1 (resp. Example 3.1) we write π as a composition of tranpo-sitions. This time π = σ ◦ τ ◦ τ , with τ := ( i x i p ) , τ := ( i y i x )and σ being the composition of the remaining transpositions. i p and i x are fixed points of σ , as τ ◦ τ ( i p ) = π ( i p ) and τ ◦ τ ( i x ) = π ( i x ). So σ permutes p − AXIMAL NON-EXCHANGEABILITY IN DIMENSION D τ ◦ τ = ( i y i x i p ), with Example 4.1 we get | C ( u ) − C ( u π ) | ≤ | C ( u ) − C ( u σ ) | + | C ( u σ ) − C ( u π ) |≤ p − X k = ⌈ p − ⌉ +1 , k = x ( u i k − u i ) + ( u i p − u i y ) ≤ p X k = ⌈ p ⌉ +1 ( u i k − u i )by the induction hypothesis and Lemma 3.1. Case . x = y : With π = σ ◦ τ and τ := ( i x i p ) (see Case 1 or Example 3.1), we get (15)analogous to Case 1. Case . x < y : Similar to case 3.1. we write π = σ ◦ τ ◦ τ . This time with τ := ( i x i y ) , τ := ( i y i p )we get (15) analogous to Case 3.1. Case . y ∈ (cid:8) , . . . , (cid:6) p (cid:7) − (cid:9) : With i x := π − ( i p ) this case can be solved analogous to Case 3,which completes the proof. (cid:3) Now we are able to prove, that for d >
2, the point u ∗ , where maximal non-exchangeability ispossible, is unique if and only if the dimension is odd. Lemma 4.2.
Let d > , C d := { C : [0 , d → R : C is a d -copula } and M := n u ∈ [0 , d : u ≤ . . . ≤ u d , ∃ π ∈ S d ∃ C ∈ C d s.t. | C ( u ) − C ( u π ) | = d − d +1 o .Then |M| = 1 if and only if d = 2 n +1 (for a n ∈ N ). If d = 2 n , then M is a ( n − -dimensionalmanifold.Proof. Let u ∈ M and ˜ u i := u i − d − d +1 ∈ (cid:2) , d +1 (cid:3) ( ∗ ). The left bound of ˜ u i follows from d − d +1 = | C ( u ) − C ( u π ) | ≤ M d ( u ) ≤ u i for any i = 1 , . . . , d . From (8) we find that any such u satisfies 2 d − d +1 − P di =2 ˜ u i ≥ d − d +1 , i.e., it holds that d − d + 1 ≥ d X i =2 ˜ u i ≥ d X i = ⌈ d ⌉ +1 ˜ u i . This and the inequality P di = ⌈ d ⌉ +1 ( u i − u ) = P di = ⌈ d ⌉ +1 (˜ u i − ˜ u ) ≥ d − d +1 from Lemma 4.1 yield(16) d X i = ⌈ d ⌉ +1 ˜ u i = d − d + 1for every u ∈ M . Let d = 2 n + 1 then the only way for (16) to be true is˜ u = . . . = ˜ u ⌈ d ⌉ = 0 , ˜ u ⌈ d ⌉ +1 = . . . = ˜ u d = 2 d + 1as 0 ≤ ˜ u j ≤ d +1 for all j = 1 , . . . , d .Now let d = 2 n and u ∈ [0 , d with u = . . . = u n = d − d + 1 , u n + j = dd + 1 + δ j for j = 1 , . . . , n such that δ j ∈ (cid:2) , d +1 (cid:3) and δ ≤ . . . ≤ δ n , n X j =1 δ j = n − d + 1holds. Let ˜ M be the set of all such u . For each u ∈ ˜ M there exists a permutation π and acopula C , such that | C ( u ) − C ( u π ) | = d − d +1 . We will construct such a copula by the Shuffle of Min method, presented by Mikusi´nski et al. (1992) and Durante and Fern´andez-S´anchez (2010),in the Appendix. Therefore, we have ˜
M ⊆ M . Now, let u ∈ M . If we assume u n +1 < dd +1 ,i.e., ˜ u n +1 < d +1 then equation (16) implies that there exists some ˜ u n + j > d +1 contradicting ( ∗ )in the beginning of the proof. Hence, we can write u n + j = dd +1 + δ j with 0 ≤ δ ≤ · · · ≤ δ n . Consequently we have ˜ u n + j = d +1 + δ j , j = 1 , . . . , n and equation (16) implies n X j =1 δ j = n − d + 1which means that u ∈ ˜ M and thus M ⊆ ˜ M . (cid:3) The above proof shows, that for every u ∈ M the first ⌈ d ⌉ components are equal. Therefore,even for a fixed u ∗ ∈ M and a fixed C ∈ C d there’s never a unique π ∈ S d which maximizes (3)(for d > π be such a permutation, then ˜ π := π ◦ τ maximizes (3) as well.5. Acknowledgements
The authors wish to thank the anonymous referees for their suggestions, which—among others—led to a simpler and shorter proof of Lemma 3.1.
Appendix A. Examples
Let d = 2 n and u ∈ M as described in the proof of Lemma 4.2, i.e. u = . . . = u n = d − d + 1 , u n + j = dd + 1 + δ j for j = 1 , . . . , n such that δ j ∈ (cid:2) , d +1 (cid:3) and δ ≤ . . . ≤ δ n , n X j =1 δ j = n − d + 1 .Let π ∈ S be the order reversing permutation, i.e. π ( j ) = d − j + 1 for j = 1 , . . . , d . By applyingthe Shuffle-Of-Min-Method, we will construct a copula C , such that | C ( u ) − C ( u π ) | = d − d +1 .According to Remark 2.1. in Durante and Fern´andez-S´anchez (2010), all that is needed for theconstruction of such a copula, is a so called shuffling structure of d -dimensional orthotopes anda system of copulas ( C i ). We use C i ≡ M d for all i for simplicity, but other choices, especiallynon-singular copulas are possible. Now for the orthotopes J i × . . . × J di (with i ∈ { , . . . , n − } ):In the following, we will give J ki for all cases of i ∈ { , . . . , n − } and k ∈ { , . . . , d } . Case . i ∈ { , . . . , n − } : Case . k ∈ { , . . . , n − i } ∪ { n + 1 , . . . , n } then: J ki := (cid:2)P i − j =1 (cid:0) d +1 + δ j (cid:1) , P ij =1 (cid:0) d +1 + δ j (cid:1)(cid:3) Case . k = n − i + 1 then: J ki := (cid:2) d − d +1 , dd +1 + δ i (cid:3) Case . i ≥ k ∈ { n − i + 2 , . . . , n } then: J ki := (cid:2)P i − j =1 ,j = n +1 − k (cid:0) d +1 + δ j (cid:1) , P ij =1 ,j = n +1 − k (cid:0) d +1 + δ j (cid:1)(cid:3) Case . i ∈ { n, . . . , n − } and: Case . k = 1 then: J ki := (cid:2) d − d +1 − δ n + P i − nj =1 (cid:0) d +1 − δ j (cid:1) , d − d +1 − δ n + P i − n +1 j =1 (cid:0) d +1 − δ j (cid:1)(cid:3) Case . n ≥ k ∈ { , . . . , n − i − } then: J ki := (cid:2) d − d +1 − δ n − δ n +1 − k + P i − nj =1 (cid:0) d +1 − δ j (cid:1) , d − d +1 − δ n − δ n +1 − k + P i − nj =1 (cid:0) d +1 − δ j (cid:1)(cid:3) Case . k = 2 n − i then: J ki := (cid:2) dd +1 + δ n +1 − k , (cid:3) AXIMAL NON-EXCHANGEABILITY IN DIMENSION D Case . i ≥ n + 1 and k ∈ { n − i + 1 , . . . , n } then: J ki := (cid:2) d − d +1 − δ n + P i − nj =1 (cid:0) d +1 − δ j (cid:1) , d − d +1 − δ n + P i − n +1 j =1 (cid:0) d +1 − δ j (cid:1)(cid:3) Case . k ∈ { n + 1 , . . . , n } then: J ki := (cid:2) dd +1 − δ n + P i − nj =1 (cid:0) d +1 − δ j (cid:1) , dd +1 − δ n + P i − n +1 j =1 (cid:0) d +1 − δ j (cid:1)(cid:3) Case . i ∈ { n − , . . . , n − } and: Case . k = 1 then: J ki := (cid:2) d − d +1 + P i − n +1 j =1 (cid:0) d +1 − δ j (cid:1) , d − d +1 + P i − n +2 j =1 (cid:0) d +1 − δ j (cid:1)(cid:3) Case . k ∈ { , . . . , n } then: J ki := (cid:2) d − d +1 + P i − n +1 j =1 (cid:0) d +1 − δ j (cid:1) , d − d +1 + P i − n +2 j =1 (cid:0) d +1 − δ j (cid:1)(cid:3) Case . k ∈ { n + 1 , . . . , n − } \ { i − n + 2 } then: J ki := (cid:2) dd +1 + P i − n +1 j =1 ,j = k − n (cid:0) d +1 − δ j (cid:1) , dd +1 + P i − n +2 j =1 ,j = k − n (cid:0) d +1 − δ j (cid:1)(cid:3) Case . k = i − n + 2: J ki := (cid:2) dd +1 + δ k − n , (cid:3) Case . i = 3 n − Case . k = 1: J ki := (cid:2) d − d +1 , (cid:3) Case . k ∈ { , . . . , n } then: J ki := (cid:2) d − d +1 , d − d +1 (cid:3) Case . k ∈ { n + 1 , . . . , n } then: J ki := (cid:2) d − d +1 − δ n , dd +1 − δ n (cid:3) By Definition 2.1. in Durante and Fern´andez-S´anchez (2010), the intervals J ki must fulfill fourconditions, in order to get a proper copula:(1) First, i must run in a finite or countable index set. This is obviously the case, as1 ≤ i ≤ n − k ∈ { , . . . , d } and i = i the intervals J ki and J ki have at most oneendpoint in common. This condition is tedious to verify, but nonetheless fulfilled.(3) Third, the orthotopes must be d -hypercubes, i.e. (cid:12)(cid:12) J k i (cid:12)(cid:12) = (cid:12)(cid:12) J k i (cid:12)(cid:12) for every i and every pair k , k . This is the case, as for every k (cid:12)(cid:12) J ki (cid:12)(cid:12) = d +1 + δ i for i ∈ { , . . . , n − } , d +1 + δ i − n +1 for i ∈ { n, . . . , n − } , d +1 + δ i − n +2 for i ∈ { n − , . . . , n − } , d +1 for i = 3 n − k , the length of the intervals must sum up to 1. n − X i =1 (cid:12)(cid:12) J ki (cid:12)(cid:12) = n − X i =1 (cid:0) d +1 + δ i (cid:1) + n − X i = n (cid:0) d +1 − δ i − n +1 (cid:1) + n − X i =2 n − (cid:0) d +1 − δ i − n +2 (cid:1) + d +1 = 1for every k .Analogous to (13) we get an explicit expression of C , namely(17) C ( u ) = n − X i =1 min (cid:16)(cid:0) u − a i (cid:1) + , . . . , (cid:0) u d − a di (cid:1) + , (cid:12)(cid:12) J i (cid:12)(cid:12)(cid:17) where a ki is the left limit of the interval J ki . The distribution of mass within the d -hypercubesis arbitrary, as long as there is exactly the mass (cid:12)(cid:12) J i (cid:12)(cid:12) in the hypercube J i × . . . × J di . In ourexample, all the mass is on the diagonal. For a non-singular copula, one could spread the massevenly within the hypercubes, for example replace M d in (13) by the Independence Copula π d .Let’s clarify things with two small examples for d = 4: Example A.1.
In (9) we get u ∗ = (0 . , . , . , C ∗ ( u ) = min (cid:8) ( u − . + , ( u − . + , u , u (cid:9) ++ min (cid:8) ( u − . + , ( u − . + , ( u − . + , u (cid:9) + min (cid:8) ( u − . + , ( u − . + , ( u − . + , u (cid:9) + min (cid:8) ( u − + | {z } =0 , . . . (cid:9) .Therefore, we have | C ∗ ( u ∗ ) − C ∗ (1 , . , . , . | = 0 . Example A.2.
As the dimension is even, the point u ∗ in Example A.1 is not the only one, inwhich maximal non-exchangeability is achieved. Let δ ∈ [0 , .
1] and ˜ u = (0 . , . , . δ , − δ ).Note that 1 − δ = 0 . δ if δ + δ = 0 .
2. The copula in (17) is given by C ( u ) = min (cid:8) u , ( u − . + , u , u , . δ (cid:9) ++ min (cid:8) ( u − . − δ ) + , ( u − . δ ) + , ( u − . − δ ) + , ( u − . − δ ) + , . δ (cid:9) + min (cid:8) ( u − . + , u , ( u − . − δ ) + , ( u − . + , . − δ (cid:9) + min (cid:8) ( u − . δ ) + , ( u − . δ ) + , ( u − . + , ( u − δ ) + , δ (cid:9) + min (cid:8) ( u − . + , ( u − . + , ( u − . − δ ) + , ( u − . − δ ) + , . (cid:9) .Therefore, we have | C ( ˜ u ) − C (1 − δ , . δ , . , . | = 0 .
6. This copula C is different fromthe copula C ∗ in Example A.1, as C ( ˜ u ) = 0, but C ∗ ( ˜ u ) = δ . References
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