On attainability of Kendall's tau matrices and concordance signatures
OOn attainability of Kendall’s tau matricesand concordance signatures
A.J. MCNEIL *1 , J.G. NE ˇSLEHOV ´A , and A.D. SMITH The York Management School, University of York, UK Department of Mathematics and Statistics, McGill University, Montr´eal, Canada University College Dublin, IrelandNovember 25, 2020
Abstract
The concordance signature of a multivariate continuous distribution is the vector of concordanceprobabilities for margins of all orders; it underlies the bivariate and multivariate Kendall’s tau measureof concordance. It is shown that every attainable concordance signature is equal to the concordancesignature of a unique mixture of the extremal copulas, that is the copulas with extremal correlationmatrices consisting exclusively of ’s and − ’s. This result establishes that the set of attainable Kendallrank correlation matrices of multivariate continuous distributions in arbitrary dimension is the set ofconvex combinations of extremal correlation matrices, a set known as the cut polytope. A methodologyfor testing the attainability of concordance signatures using linear optimization and convex analysis isprovided. The elliptical copulas are shown to yield a strict subset of the attainable concordance signaturesas well as a strict subset of the attainable Kendall rank correlation matrices; the Student t copula isseen to converge to a mixture of extremal copulas sharing its concordance signature with all ellipticaldistributions that have the same correlation matrix. A method of estimating an attainable concordancesignature from data is derived and shown to correspond to using standard estimates of Kendall’s tau inthe absence of ties. The methodology has application to Monte Carlo simulations of dependent randomvariables as well as expert elicitation of consistent systems of Kendall’s tau dependence measures. Keywords : Attainable correlations; Concordance; Copulas; Cut-polytope; Elliptical distributions; Extremaldistributions; Kendall’s rank correlation; Multivariate Bernoulli distributions.
Kendall’s tau is a well-known and widely used measure of association. The sample version is usually at-tributed to Kendall (1938) although both the sample coefficient and its population analogue can be discernedin the work of earlier authors, according to Kruskal (1958). As an ordinal or margin-free measure of depen-dence, Kendall’s tau has a number of advantages over Pearson’s linear correlation, which make it appealingin a wide variety of practical applications (see Embrechts et al., 2002, for example). * Address correspondence to Alexander J. McNeil, The York Management School, University of York, Freboys Lane, YorkYO10 5GD, UK, +44 (0) 1904 325307, [email protected] . a r X i v : . [ m a t h . S T ] N ov −1.0 −0.5 0.0 0.5 1.0 − . − . . . . {1,3} { , } −1.0 −0.5 0.0 0.5 1.0 − . − . . . . {1,3} { , } Figure 1: The cut polytope of possible values of τ { , } , τ { , } , and τ { , } in dimension d = 3 (left panel).The range of possible values of τ { , } and τ { , } when τ { , } is fixed at − . (middle panel) and (rightpanel).For a random vector X = ( X , . . . , X d ) with continuous margins, the population value of Kendall’s taufor two components i, j ∈ { , . . . , d } is given by τ { i,j } = τ ( X i , X j ) = P (cid:0) ( X i − X ∗ i )( X j − X ∗ j ) > (cid:1) − P (cid:0) ( X i − X ∗ i )( X j − X ∗ j ) < (cid:1) , where X ∗ is a random vector independent of X with identical distribution. The matrix P τ of pairwisetau values is termed the (population) Kendall rank correlation matrix. If the margins F , . . . , F d of X arecontinuous, as we shall assume throughout this paper, P τ depends only on the unique underlying copula of X . The latter is the distribution function (df) C of the random vector U = ( F ( X ) , . . . , F d ( X d )) (seeNelsen, 2006, for example). Thus the entries of P τ do not depend on the marginal distributions of X andthe off-diagonal elements satisfy the axioms of a measure of concordance proposed by Scarsini (1984).While Kendall’s tau has been widely studied in the bivariate case, the exact conditions that characterizethe set of d × d Kendall rank correlation matrices have not been fully understood. In this paper we prove aconjecture of Hofert and Koike (2019) that the set of attainable Kendall rank correlation matrices is identicalto the set of convex combinations of the extremal correlation matrices, that is, the correlation matrices con-sisting exclusively of ’s and − ’s; this set is also known as the cut polytope (Laurent and Poljak, 1995). Ourapproach relies on so-called mixtures of extremal copulas, that is mixtures of the d − possible copulas withextremal correlation matrices, as studied by Tiit (1996). We also make explicit the links between extremalmixture copulas and multivariate Bernoulli distributions, and explain why the set of attainable Kendall cor-relation matrices is identical to the set of attainable correlation matrices for multivariate Bernoulli randomvectors with symmetric margins as derived by Huber and Mari´c (2015, 2019).The cut polytope of attainable values of τ { , } , τ { , } , and τ { , } in dimension d = 3 is shown in theleft panel Figure 1. It is apparent from it that it is strictly smaller than [ − , . This phenomenon is evenmore apparent in the middle and right panel of the same figure, which show the set possible values of τ { , } and τ { , } when τ { , } is fixed at − . and respectively. The value of τ { , } restricts the possible valuesof τ { , } and τ { , } . The results derived in this paper offer a method of checking the logical coherenceof systems of pairwise Kendall’s tau rank correlations and a means of finding a priori bounds on the newpairwise correlations that are introduced when additional variables are added to the system.The characterization of Kendall rank correlation matrices contributes to the growing literature on com-patibility and attainability problems for multivariate dependence measures. Pearson correlation matrices areperhaps the most widely studied and the so-called elliptope of positive semi-definite correlation matrices hasbeen investigated by Huber and Mari´c (2015, 2019), among others. Hofert and Koike (2019) have shown2hat the set of Blomqvist’s beta matrices is the cut polytope while Embrechts et al. (2016) found conditionscharacterizing tail dependence matrices. The work of Devroye and Letac (2015) and Wang et al. (2019)considers the set of attainable Spearman rank correlation matrices; this coincides with the elliptope of linearcorrelation matrices when d (cid:54) but not when d (cid:62) , while the case d ∈ { , } remains unresolved.Our results for Kendall rank correlation matrices are in fact part of a bigger picture. Going beyondpairwise dependence, higher-order measures of association for the vector X can be defined in terms ofprobabilities of concordance κ I for sub-vectors X I corresponding to subsets I ⊆ { , . . . , d } of the indices.The probabilities of concordance are given in general by κ I = κ ( X I ) = 2 P ( X I < X ∗ I ) = P ( { X I < X ∗ I } ∪ { X ∗ I < X I } ) , (1)where X ∗ again denotes an independent random vector with the same distribution as X and { X I < X ∗ I } = { X i < X ∗ i , i ∈ I } . The bivariate probabilities κ { i,j } are related to the Kendall’s tau rank correlation values τ { i,j } by κ { i,j } − τ { i,j } and higher-dimensional concordance probabilities are related to a higher-dimensional generalization of Kendall’s tau proposed by Joe (1990) and studied by Genest et al. (2011). Weintroduce the term concordance signature of X to describe the vector ( κ I : I ∈ P ( D )) where P ( D ) isthe power set of the index set D = { , . . . , d } . As in the bivariate case, the probabilities κ I and hence theconcordance signature are determined by the unique copula C of X .Our main result is to prove that the concordance signature of a copula C in dimension d is always equalto the concordance signature of a unique mixture of the extremal copulas. This insight allows us to identifythe set of all possible concordance signatures of d -dimensional random vectors with continuous marginsas a convex polytope and to derive a method for testing the attainability of concordance signatures. Themethodology we propose has a number of important applications.First, it sheds light on the difficult problem of compatibility of lower-dimensional margins of higher-dimensional copulas, as considered by Joe (1996, 1997) among others; the compatibility of the concordanceprobabilities implied by the lower-dimensional margins provides a necessary condition if they are to formthe margins of a higher-dimensional copula. We can use our result to work out whether a particular set ofputative concordance probabilities forms a subset of the concordance signature of a copula and, if it does,to determine possible values for the remainder of the signature. The solution involves the application ofstandard techniques from linear optimization and convex analysis.Second, our methodology allows us to take a close look at the dependence structure inherent in thewidely used class of elliptical copulas. We show that the concordance signatures of the family of ellipticalcopulas form a strict subset of the set of all possible attainable concordance signatures. The surprisingconsequence is the existence of Kendall correlation matrices that do not correspond to elliptically distributedrandom vectors; this behaviour provides a contrast with Pearson correlation matrices, which are always thecorrelation matrices of elliptical distributions. Moreover, we show that the d -dimensional Student t copulawith correlation matrix P ∈ R d × d and degree-of-freedom parameter ν > converges point-wise to amixture of extremal copulas as ν → which shares its concordance signature. Thus the t copulas with ν < provide a class of absolutely continuous distributions that can model behaviour that is close to thatof a subclass of the mixtures of extremal copulas, namely the subclass sharing concordance signatures withthe elliptical copulas.Finally, our work has many applications to real data. We prove that estimating concordance probabilitiesusing classical estimators of bivariate and multivariate Kendall’s tau leads to intrinsic concordance signatureswhich correspond to valid copulas. This offers a powerful technique in Monte Carlo simulation studies orrisk analyses. It allows us to generate data from a model with identical concordance signature to thatestimated from the data but with a very extreme form of tail dependence in which extreme values (of largeor small size) are always coincident in each margin. Moreover, having estimated concordance probabilitiesor Kendall’s tau values from data, our methods allow us to derive bounds on the values that could be obtained3f new variables were added to the dataset. They also allow expert opinions about Kendall rank correlationsto be integrated into the estimation procedure or challenged for inconsistency. Let X = ( X , . . . , X d ) denote a generic random vector with continuous marginal distributions F , . . . , F d .As is well known, the unique copula C of X is the distribution function of the random vector U =( U , . . . , U d ) where U i = F i ( X i ) for i = 1 , . . . , d . For I ⊆ D = { , . . . , d } with I (cid:54) = ∅ we write X I and U I to denote sub-vectors of X and U and C I for the copula of X I or distribution function of U I .Let U and U ∗ be independent random vectors with distribution given by the copula C . The concordanceprobabilities (1) corresponding to subsets I ∈ P ( D ) with | I | (cid:62) can be written as κ I = κ ( C I ) = 2 P ( U I (cid:54) U ∗ I ) = 2 (cid:90) [0 , | I | C I ( u ) dC I ( u ) . (2)For the subsets I ∈ P ( D ) with | I | < we note that (1) implies κ { i } = 1 for a singleton and we adopt theconvention κ ∅ = 1 .The concordance probabilities for | I | (cid:62) are related to the multivariate Kendall’s tau coefficients τ ( U I ) analysed in Genest et al. (2011) by the formula (2 | I |− − τ ( U I ) = 2 | I |− κ ( U I ) − . (3)From Proposition 1 in Genest et al. (2011), which is based on the exclusion-inclusion principle, we candeduce that for any set I ⊆ D of odd cardinality κ I = 1 + (cid:88) A ⊂ I, (cid:54) | A | < | I | ( − | A | P ( U A (cid:54) U ∗ A ) = 1 − | I | (cid:88) A ⊂ I, (cid:54) | A | < | I | ( − | A | κ A . (4)This means that we only need the concordance probabilities for subsets of D of even cardinality to determinethe full concordance signature. That is, we only need consider the vector κ = ( κ ( C I ) : I ∈ E ( D )) where E ( D ) is the even power set defined by E ( D ) = { I : I ⊆ D , | I | even } ; note that E ( D ) contains the empty set ∅ . When considering attainability problems under partial specification of a concordance signature it willbe necessary to associate the elements of the partial concordance signature with the elements of the powerset P ( D ) to which they pertain. We introduce the notion of a label set and some further terminology in thefollowing definition. Definition 1.
1. A label set is any collection S of subsets of D such that ∅ ∈ S .2. A vector κ is a partial concordance signature of a copula C with label set S if κ = ( κ ( C I ) : I ∈ S ) .By convention, the elements of S are taken in lexicographical order.3. When the label set is E ( D ) the partial concordance signature is called the even concordance signatureof C and is denoted κ ( C ) .4. When the label set is P ( D ) the partial concordance signature is called the full concordance signatureof C and is denoted ˜ κ ( C ) . 4 Extremal mixture copulas
Our main tool for the study of attainable concordance signatures is the family of mixtures of extremalcopulas, henceforth referred to as extremal mixture copulas. In particular, we make use of the relationshipbetween extremal mixture copulas and multivariate Bernoulli distributions. In this section we develop thenecessary notation and theory.
An extremal copula C with index set J ⊆ D is the df of the random vector U = ( U , . . . , U d ) where for j ∈ D , U j d = (cid:40) U if j ∈ J , − U if j ∈ J (cid:123) ,where U is a standard uniform random variable. For all u = ( u , . . . , u d ) ∈ [0 , d it has the explicit form C ( u ) = (cid:18) min j ∈ J u j + min j ∈ J (cid:123) u j − (cid:19) + , (5)where for any x ∈ R , x + = max( x, denotes the positive part of x .In dimension d (cid:62) there are d − extremal copulas and we enumerate them in the following way. For k ∈ { , . . . , d − } let s k = ( s k, , . . . , s k,d ) be the vector consisting of the digits of k − when representedas a d -digit binary number. For example, when d = 4 we have exactly eight extremal copulas correspondingto the vectors s = (0 , , , , s = (0 , , , , s = (0 , , , , s = (0 , , , , s = (0 , , , , s = (0 , , , , s = (0 , , , , s = (0 , , , . The extremal copulas concentrate their probability mass on the diagonals of the unit hypercube between thepoints given by the vectors s k = ( s k, , . . . , s k,d ) and the vectors − s k = (1 − s k, , . . . , − s k,d ) ; thereare d − such diagonals.For the vector s k we define J k ⊆ D = { , . . . , d } to be the set of indices corresponding to zeros and J (cid:123) k to be the set of indices corresponding to ones; in other words, j ∈ J k if s k,j = 0 and j ∈ J (cid:123) k if s k,j = 1 . Forthe 4-dimensional example above we have J = { , , , } , J (cid:123) = ∅ , J = { , , } , J (cid:123) = { } and so on.The k th extremal copula in dimension d is the extremal copula with index set J k and denoted by C ( k ) . Notethat J = D so that C (1) ( u ) = min( u , . . . , u d ) is the comonotonicity or Fr´echet–Hoeffding upper boundcopula.Extremal copulas owe their name to the fact that their correlation matrixes are extremal correlationmatrices, that is, consisting exclusively of 1’s and − ’s. Indeed, for any k ∈ { , . . . , d − } , we find that theKendall correlation matrix of the k th extremal copula C ( k ) is P ( k ) = (2 s k − )(2 s k − ) (cid:62) . This correlationmatrix is simultaneously the matrix of pairwise Pearson and Spearman correlations of the random vector U ( k ) distributed as C ( k ) . We next define extremal mixture copulas which play a pivotal role in this paper.
Definition 2.
An extremal mixture copula is a copula of the form C ∗ = (cid:80) d − k =1 w k C ( k ) , where the weightssatisfy w k (cid:62) for all k ∈ { , . . . , d − } and (cid:80) d − k =1 w k = 1 .5or k ∈ { , . . . , d − } , let U ( k ) be a random vector with df C ( k ) in (5). If N is a multinomial randomvariable taking values in { , . . . , d − } and independent of U (1) , . . . , U (2 d − ) , then it is easily seen that thedistribution function of the random vector U ( N ) such that U ( N ) = U ( k ) if N = k is an extremal mixturecopula with weights w k = P ( N = k ) , k ∈ { , . . . , d − } .The following proposition shows how the extremal mixture copulas are related to multivariate Bernoullidistributions. Proposition 1.
Let U be a standard uniform random variable and B a multivariate Bernoulli vector inde-pendent of U . Then the distribution function of the vector U B + (1 − U )( − B ) (6) is an extremal mixture with weights given, for each k ∈ { , . . . , d − } , by w k = P ( B = s k ) + P ( B = − s k ) , (7) where s k is as defined in Section 3.1. Conversely, any extremal mixture copula C ∗ = (cid:80) d − k =1 w k C ( k ) is thedistribution function of a random vector of the form (6) , where U is independent of B and (7) holds for all k ∈ { , . . . , d − } .Proof. Let U be of the form (6) where U and B are independent. For any u ∈ [0 , , P ( U (cid:54) u ) = (cid:88) b ∈{ , } d P ( max j : b j =0 (1 − v j ) (cid:54) U (cid:54) min j : b j =1 v j | B = b ) P ( B = b )= (cid:88) b ∈{ , } d (cid:18) min j : b j =1 v j + min j : b j =0 v j − (cid:19) + P ( B = b )= d − (cid:88) k =1 (cid:32) min j ∈ J k v j + min j ∈ J (cid:123) k v j − (cid:33) + (cid:16) P ( B = s k ) + P ( B = − s k ) (cid:17) . where in the final step we have used the fact that the set of possible outcomes of the Bernoulli vector B canbe written as the disjoint union { , } d = d − (cid:91) k =1 { s k , − s k } . (8)From (5), one can see that the distribution function of U is indeed an extremal mixture with weights as givenin (7).Conversely, given an extremal mixture copula of the form C ∗ = (cid:80) d − k =1 w k C ( k ) , it suffices to considerany Bernoulli vector B independent of U with the property that w k = P ( B = s k ) + P ( B = − s k ) ;this is possible because the events { B = s k } and { B = − s k } are disjoint and their union formsa partition of { , } d by (8). We can then retrace the steps of the argument in reverse to establish therepresentation (6).Proposition 1 establishes that extremal mixture copulas have a stochastic representation (6). However,the class of Bernoulli distributions satisfying (7) is infinite since the mass w k can be split between the events { B = s k } and { B = − s k } in an arbitrary way. It is important for the arguments used in this paper tosingle out a representative and we do this by setting B = Y , where Y is radially symmetric about . . Thismeans that Y d = − Y and implies that, for each k ∈ { , . . . , d − } , P ( Y = s k ) = P ( Y = − s k ) = 0 . w k . (9)6uch distributions are also known as palindromic Bernoulli distributions (Marchetti and Wermuth, 2016)and they are fully parameterized by the d − probabilities P ( Y = s k ) = 0 . w k , k ∈ { , . . . , d − } . Thismeans that there is a bijective mapping between the extremal mixture copulas and the radially symmetricmultivariate Bernoulli distributions in dimension d . The vector of weights w describing the extremal mixturecopula is twice the vector of probabilities that parameterizes the multivariate Bernoulli distribution of Y .A number of constraints apply to radially symmetric Bernoulli random vectors: they must have marginalprobabilities P ( Y i = 1) = 0 . for all i ; moreover, all probabilities of the form p I = P ( Y I = ) for sets I with odd cardinality are fully determined by the equivalent probabilities for lower-dimensional sets of evencardinality. This follows from the fact that p I = E (cid:32)(cid:89) i ∈ I Y i (cid:33) = E (cid:32)(cid:89) i ∈ I (1 − Y i ) (cid:33) = 1 + (cid:88) A ⊆ I, | A | (cid:62) ( − | A | E (cid:32)(cid:89) i ∈ A Y i (cid:33) . When I is odd, we then have p I = 1 + (cid:88) A ⊂ I, (cid:54) | A | < | I | ( − | A | p A = 1 − | I | + (cid:88) A ⊂ I, (cid:54) | A | < | I | ( − | A | p A , (10)where the last equality follows from the fact that p A = 0 . whenever | A | = 1 . For example, p { , , } = E (cid:16) (1 − Y )(1 − Y )(1 − Y ) (cid:17) = 12 (cid:0) p { , } + p { , } + p { , } (cid:1) − . Since the vector of joint event probabilities ( p I : I ∈ P ( D ) \ ∅ ) uniquely specifies the distribution of aBernoulli random vector, this leads to the insight that we need only consider sets I in the even power set. Infact we have the following result. Proposition 2. (i) The radially symmetric Bernoulli distribution of a vector Y in dimension d is uniquelydetermined by the vector of probabilities p Y = ( p I : I ∈ E ( D ) \ ∅ ) .(ii) For label sets S ⊂ P ( D ) the vector p Y is the shortest vector of the form ( p I : I ∈ S \ ∅ ) whichuniquely determines the distribution of Y for all possible Y .Proof. For part (ii) observe that the vector p Y has length equal to (cid:98) d/ (cid:99) (cid:88) j =1 (cid:18) d j (cid:19) = 2 d − − . Radially symmetric multivariate Bernoulli distributions in dimension d have d − − free parameters, withone being deducted for the sum constraint (cid:80) d − k =1 P ( Y = s k ) = 1 . Thus the vector p Y is the minimalvector of its kind that is required to fully specify the distribution of Y in all cases.We close this section by noting that the independence between U and Y in the stochastic representation U d = U Y + (1 − U )( − Y ) of an extremal mixture copula is an important assumption. An example of acopula that violates this assumption is provided in Appendix A and reveals an interesting contrast betweenthe extremal copulas and the mixtures of extremal copulas. While a necessary and sufficient condition for avector U to be distributed according to an extremal copula is that its bivariate margins should be extremalcopulas, the analogous statement does not hold for extremal mixture copulas. Although it is necessary thatthe bivariate margins of an extremal mixture copula are extremal mixture copulas, the example shows thatthis is not sufficient. An additional condition is required, as stated below and proved in Appendix A. Proposition 3.
The distribution of a random vector U = ( U , . . . , U d ) is a mixture of extremal copulas ifand only if its bivariate marginal distributions are mixtures of extremal copulas and for all u ∈ [0 , , P (cid:16) U (cid:54) u | I { U j = U } , j (cid:54) = 1 (cid:17) = u . (11)7 Main result
In this section we characterize concordance signatures of arbitrary copulas in dimension d . To do so we firstcalculate concordance signatures of extremal mixture copulas and investigate their properties.For k = 1 , . . . , d − , let X ( k ) = ( X ( k )1 , . . . , X ( k ) d ) be a random vector with continuous margins andcopula C ( k ) . Then the sets { X ( k ) j : j ∈ J k } and { X ( k ) j : j ∈ J (cid:123) k } are sets of comonotonic random variables,while any pair ( X ( k ) i , X ( k ) j ) such that i ∈ J k and j ∈ J (cid:123) k , or vice versa, is a pair of countermonotonicrandom variables. For I ⊆ D we introduce the notation a I,k = (cid:40) if I ⊆ J k or I ⊆ J (cid:123) k , otherwise. (12)For sets with | I | > the value a I,k indicates whether the components of X ( k ) with indices in I are comono-tonic variables or not. However the values a I,k are also defined for sets I which are singletons and for theempty set; in these cases we have a { i } ,k = 1 and a ∅ ,k = 1 for all k and i . Proposition 4. (i) For any k ∈ { , . . . , d − } and I ∈ P ( D ) , κ ( C ( k ) I ) = a I,k .(ii) If C is an extremal mixture copula of the form (cid:80) d − k =1 w k C ( k ) then, for any I ⊆ D , κ ( C I ) = d − (cid:88) k =1 w k κ (cid:16) C ( k ) I (cid:17) = d − (cid:88) k =1 w k a I,k . (13) Proof.
Statement (i) follows easily by observing that, if the variables indexed by I are comonotonic, thenthey are concordant with probability one, otherwise they are disconcordant; if I is a singleton or the emptyset the formula still holds.To establish (ii), note again that (13) holds trivially for sets I which are singletons or the empty set. Forsets such that | I | (cid:62) we can use (2) to write κ ( C ) = 2 (cid:90) [0 , d C ( u ) dC ( u ) = 2 d − (cid:88) j =1 2 d − (cid:88) k =1 w j w k (cid:90) [0 , d C ( j ) ( u ) dC ( k ) ( u ) . Introducing independent random vectors U ( j ) ∼ C ( j ) and ˜ U ( k ) ∼ C ( k ) for j ∈ { , . . . , d − } and k ∈{ , . . . , d − } we calculate that (cid:90) [0 , d C ( j ) ( u ) dC ( k ) ( u ) = P (cid:16) U ( j ) (cid:54) ˜ U ( k ) (cid:17) = if j = k = 1 , if j = 1 or k = 1 but j (cid:54) = k , if j (cid:54) = 1 and k (cid:54) = 1 .Hence we can verify that κ ( C ) = 2 w w (cid:16) w · · · + w d − (cid:17) = w + w (1 − w ) = w , which is the weight on C (1) = M , the d -dimensional comonotonicity copula. If U is distributed accordingto C then the vector U I is distributed according to a mixture of extremal copulas in dimension | I | and it8ollows that κ ( U I ) = κ ( C I ) = ˜ w where ˜ w is the weight attached to the case where U I is a comonotonicrandom vector. This is given by ˜ w = d − (cid:88) k =1 w k a I,k = d − (cid:88) k =1 w k κ (cid:16) C ( k ) I (cid:17) , where the second equality follows from part (i).A simple consequence of Proposition 4, specifically of (13) and (3), is a formula for the multivariateKendall’s tau coefficient for mixtures of extremal distributions. Corollary 1. If C = (cid:80) d − k =1 w k C ( k ) then for any I ⊆ D with | I | (cid:62) , the multivariate Kendall’s taucoefficient satisfies τ ( C I ) = d − (cid:88) k =1 w k τ (cid:16) C ( k ) I (cid:17) . (14)A further simple consequence of Proposition 4 is that we can write down the concordance signatures ofextremal copulas and extremal mixture copulas. For k ∈ { , . . . , d − } the full concordance signature of C ( k ) is ˜ κ ( C ( k ) ) = ˜ a k = ( a I,k : I ∈ P ( D )) and the full concordance signature of C = (cid:80) dk =1 w k C ( k ) is ˜ κ ( C ) = (cid:80) dk =1 w k ˜ a k .Since full concordance signatures are determined by even concordance signatures we now focus on thelatter and prove two key properties of the even concordance signature of an extremal mixture copula thatlink back to the Bernoulli representation in Section 3.2. Proposition 5.
Let a k = ( a I,k : I ∈ E ( D )) , for k ∈ { , . . . , d − } , and let κ ( C ) = (cid:80) d − k =1 w k a k be theeven concordance signature of the extremal mixture copula C .(i) κ ( C ) uniquely determines C . Moreover, it is the minimal partial concordance signature of C whichuniquely determines C in all cases.(ii) The vectors a k , k ∈ { , . . . , d − } are linearly independent.Proof. For part (i) recall that a random vector U ∼ C has the stochastic representation U d = U Y + (1 − U )( − Y ) where Y is a random vector with a radially symmetric Bernoulli distribution. For any set I ∈ E ( D ) \ ∅ the components of U Y + (1 − U )( − Y ) are concordant if and only if Y I = or Y I = .It follows from the radial symmetry property that κ I ( C ) = 2 P ( Y I = ) . By Proposition 2 the vector ( P ( Y I = ) : I ∈ E ( D ) \ ∅ ) is the minimal vector of event probabilities that pins down the law of Y in allcases and hence, by equation (9), the weights w k = 2 P ( Y = s k ) in the representation C = (cid:80) d − k =1 w k C ( k ) .If another extremal mixture copula ˜ C shares the same values for the concordance probabilities of even order,then the weights must be identical.For part (ii) suppose, on the contrary, that the vectors a k are linearly dependent, that is, there existscalars µ k , k ∈ { , . . . , d − } such that µ k (cid:54) = 0 for at least one k and d − (cid:88) k =1 µ k a k = . (15)To prove the result, we need to show that this assumption leads to a contradiction. We select a copula C such that C = (cid:80) d − k =1 w k C ( k ) and w k > for all k . Because the first component of a k is 1 for all k , we9ave that (cid:80) d − k =1 µ k = 0 . Hence, there exists at least one k ∗ such that µ k ∗ > . From (15) we have that foreach α ∈ R , (cid:80) d − k =1 ( w k − αµ k ) a k = κ ( C ) .Let α ∗ = min(( w k /µ k ) : µ k > . Clearly, α ∗ > because the set over which the minimum is takencontains at least w k ∗ /µ k ∗ and because w k > for all k . Now define w ∗ = w − α ∗ µ . These weights arenon-negative and sum up to one, while w ∗ (cid:54) = w . This implies that κ ( C ) = (cid:80) d − k =1 w k a k = (cid:80) d − k =1 w ∗ k a k but this is not possible because the mixture weights are unique by part (i).Proposition 5 shows that the set of attainable even concordance signatures of d -dimensional extremalmixture copulas is the convex hull K = d − (cid:88) k =1 w k a k : w k (cid:62) , k = 1 , . . . , d − , d − (cid:88) k =1 w k = 1 . and implies in particular that K is a convex polytope with vertices a k , k ∈ { , . . . , d − } . Similarly theset of full concordance signatures is the convex polytope with vertices ˜ a k , k ∈ { , . . . , d − } . We can nowstate and prove the main result of this paper which shows that these sets are also the sets of attainable evenand full concordance signatures for any d -dimensional copula. Theorem 1.
Let C be a d -dimensional copula and κ ( C ) = ( κ I : I ∈ E ( D )) its even concordance signature.Then there exists a unique extremal mixture copula C ∗ = (cid:80) d − k =1 w k C ( k ) such that κ ( C ) = κ ( C ∗ ) . Theweights w k are given by the unique solution of the linear equation system κ ( C ) = d − (cid:88) k =1 w k a k (16) where a k = ( a I,k : I ∈ E ( D )) for k ∈ { , . . . , d − } .Proof. For a vector U ∼ C we can write, for any I ∈ P ( D ) with | I | (cid:62) , κ I = κ ( C I ) = 2 P ( U I < U ∗ I ) = P ( U I < U ∗ I ) + P ( U I > U ∗ I )= P (cid:16) sign( U ∗ I − U I ) = (cid:17) + P (cid:16) sign( U ∗ I − U I ) = − (cid:17) where U ∗ is an independent copy of U . If we define the random vectors V = sign( U ∗ − U ) , Y = 12 ( V + ) , then the concordance probabilities of C are given by κ I = P ( Y I = ) + P ( Y I = ) (17)so that the concordance signature of C is determined by the distribution of Y , which is a radially symmetricBernoulli distribution. The radial symmetry follows from the fact that V d = − V so it must be the case that Y − d = − Y or, equivalently, Y d = − Y .From (17) we can conclude that, for every I ∈ P ( D ) , κ I = d − (cid:88) k =1 (cid:16) P ( Y = s k ) + P ( Y = − s k ) (cid:17) I { I ⊆ J k or I ⊆ J (cid:123) k } = d − (cid:88) k =1 w k a I,k = κ I ( C ∗ ) , { Y = s k } and { Y = − s k } forms apartition of { , } d in the first equality, the notation (12) and (9) in the second and (13) in the final equality.Thus the weights w k specifying the extremal mixture copula are precisely the probabilities that specify thelaw of the radially symmetric Bernoulli vector Y through w k = 2 P ( Y = s k ) and the uniqueness of theset of weights follows from the uniqueness of the law of Y . The sufficiency of solving the linear equationsystem (16) to determine the weights w k and the existence of a unique solution follows from Proposition 5,in particular the fact that (16) is a system of d − equations with d − unknowns and the vectors a k arelinearly independent.The implication of Theorem 1 is that we can find the mixture weights w for a a given concordancesignature using simple linear algebra. To see this, let κ = κ ( C ) denote the even concordance signature ofa d -dimensional copula C and let A d be the d − × d − -matrix with columns a k . Proposition 5 impliesthat A d is of full rank, and hence invertible. The linear equation system (16) can be written in the form κ = A d w and must have a unique solution which can be found by calculating w = A − d κ . For example,when d = 4 we would have κ (cid:122) (cid:125)(cid:124) (cid:123) κ { , } κ { , } κ { , } κ { , } κ { , } κ { , } κ { , , , } = A (cid:122) (cid:125)(cid:124) (cid:123) w (18)The fact that the even concordance signature is required to determine the weights of the extremal mixturecopula in all possible cases allows us to view the even concordance signature of a copula as the minimal com-plete concordance signature. Indeed, the fact that A d is of full rank has the following interesting corollary,namely that a formula such as (4) can only hold for sets of odd cardinality. Corollary 2.
When I is a set of even cardinality there is no linear formula valid for all copulas relating κ I to concordance probabilities of order | I (cid:48) | (cid:54) | I | .Proof. If the statement were not true, the rows of the matrix A d would be linearly dependent, contradictingthe assertion that A d is of full rank. Remark 1.
The implications of Corollary 2 can be extended. In view of (3) an analogous result could bestated for the multivariate Kendall’s tau coefficients: when | I | is odd, a linear formula relating τ I to thevalues for lower-dimensional subsets exists (see Proposition 1 in Genest et al. (2011)) but no such formulaexists when | I | is even. Moreover, as we will discuss in Section 6, the concordance probabilities of ellipticalcopulas are equal to twice the orthant probabilities of Gaussian distributions centred at the origin: thus if Z ∼ N d ( , Σ) is a Gaussian random vector, recursive linear formulas exist for the probabilities P ( Z I (cid:62) ) when | I | is odd but not when | I | is even. Let S ⊆ E ( D ) be a label set. Suppose we want to test whether a vector of putative concordance probabilities κ = ( κ I : I ∈ S ) with label set S could be a partial concordance signature of some copula C . The vector11 is is said to be attainable if there exists a d -dimensional copula C such that κ ( C I ) = κ I for each I ∈ S .From Theorem 1, it suffices to determine whether there exists an extremal mixture copula with this property.A particular special case of interest is when the label set of κ is the empty set together with all the setsof cardinality 2 in d dimensions:, i.e. S = {∅ , { , } , . . . , { , d } , . . . , { d − , d }} . In view of (3), this isequivalent to determining whether a matrix of putative Kendall rank correlations is attainable. For a randomvector X we denote the Kendall rank correlation matrix and the linear correlation matrix by P τ ( X ) and P ( X ) , respectively. Note that Kendall rank correlation matrices must be correlation matrices, i.e. positivesemi-definite matrices with ones on the diagonal and all elements in [ − , . This follows from the fact thatif U is a random vector distributed as a copula C , then we can write P τ ( U ) = P (sign( U − U ∗ )) , in otherwords, as the linear correlation matrix of the sign vector of the difference between U and an independentcopy U ∗ .In Theorem 2 below we provide an answer to the attainability question for Kendall rank correlationmatrices. The proof is a simple consequence of Theorem 1 and the lemma below. Lemma 1.
Let U be distributed according to the mixture of extremal copulas given by C ∗ = (cid:80) d − k =1 w k C ( k ) .Then P τ ( U ) = P ( U ) = d − (cid:88) k =1 w k P ( k ) . (19) Proof.
The equality P τ ( U ) = (cid:80) d − k =1 w k P τ ( U ( k ) ) follows from (14) while the corresponding equality P ( U ) = (cid:80) d − k =1 w k P ( U ( k ) ) for linear correlation matrices was proved by Tiit (1996). Hence (19) followsfrom the equality of linear and Kendall correlations for extremal copulas. Theorem 2.
The d × d correlation matrix P is a Kendall rank correlation matrix if and only if P can berepresented as a convex combination of the extremal correlation matrices in dimension d , that is, P = d − (cid:88) k =1 w k P ( k ) . (20) Proof. If P is of the form (20) then Lemma 1 shows that it is the Kendall’s tau matrix of the extremal mixturecopula C ∗ = (cid:80) d − k =1 w k C ( k ) . Conversely if P is the Kendall’s τ matrix of an arbitrary copula C then, byTheorem 1, it is also the Kendall’s tau matrix of the extremal mixture copula with the same concordancesignature and must take the form (19) by Lemma 1.The set of convex combinations of extremal correlation matrices is known as the cut polytope; an illus-tration for d = 3 is shown in the left panel of Figure 1. The right panels show the attainable values of τ { , } and τ { , } when τ { , } is fixed at certain values, or equivalently, when the partial concordance signature κ = (1 , κ { , } ) is fixed; in other words, a section has been taken through the polytope.Laurent and Poljak (1995) showed that the cut polytope is a strict subset of the so-called elliptope ofcorrelation matrices in dimensions d (cid:62) ; see also Section 3.3 of Hofert and Koike (2019). For example,the positive-definite correlation matrix − − − − − − is in the elliptope but not the cut polytope and therefore cannot be a matrix of Kendall’s tau values. If τ = − / and we set κ = (1 + τ ) / and κ = (1 , κ, κ, κ ) , there is no solution to the equation A w = κ on the 4-dimensional unit simplex and a representation of the form (20) is impossible.12uber and Mari´c (2019) have shown that the cut polytope is also the set of attainable linear correlationmatrices for multivariate distributions with symmetric Bernoulli marginal distributions. This can be deducedfrom Theorem 1 and Proposition 1 using the following lemma. Lemma 2.
Let U = U B + (1 − U )( − B ) where B is a Bernoulli vector with symmetric marginaldistributions and U is a uniform random variable independent of B . Then P τ ( U ) = P ( B ) .Proof. First note that P τ ( U ) = P ( U ) by Proposition 1 and Lemma 1. By writing U U (cid:62) = U BB (cid:62) + (1 − U ) ( − B )( − B ) (cid:62) + U (1 − U ) (cid:16) B ( − B ) (cid:62) + ( − B ) B (cid:62) (cid:17) = (2 U − BB (cid:62) + (2 U − − U ) (cid:0) B (cid:62) + B (cid:62) (cid:1) + (1 − U ) (cid:62) and using the fact that E ( B i ) = 0 . for all i ∈ { , . . . , d } , we find that cov( U ) = 13 (cid:18) E ( BB (cid:62) ) − (cid:16) E ( B (cid:62) ) + E ( B (cid:62) ) (cid:17) + (cid:62) (cid:19) − (cid:62) = 13 E ( BB (cid:62) ) − (cid:62) . Since var( U i ) = 1 / and var( B i ) = 1 / for all i ∈ { , . . . , d } we conclude that P ( U ) = 4 E ( BB (cid:62) ) − (cid:62) = P ( B ) . This completes the proof.
Proposition 6.
The set of Kendall rank correlation matrices of copulas is identical to the set of linearcorrelation matrices of Bernoulli random vectors with symmetric margins.Proof. If ρ τ ( U ) is a Kendall rank correlation matrix then, by Theorem 1, it is identical to the Kendall rankcorrelation matrix of a random vector U ∗ with distribution given by the extremal mixture copula with thesame concordance signature as U . It follows from Proposition 1 that U ∗ has the stochastic representation U ∗ d = U Y + (1 − U )( − Y ) where, without loss of generality, Y has a radially symmetric Bernoullidistribution (with symmetric margins). Lemma 2 gives P τ ( U ) = P τ ( U ∗ ) = P ( Y ) .Conversely if P ( B ) is the correlation matrix of a Bernoulli random vector B with symmetric margins(not necessarily radially symmetric) then by Proposition 1 we can take an independent uniform randomvariable U and construct a random vector U ∗ = U B + (1 − U )( − B ) with an extremal mixture copula.Lemma 2 gives P ( B ) = P τ ( U ∗ ) . Remark 2.
It is clear from the proof that the set of linear correlation matrices of Bernoulli random vectorswith symmetric margins is equal to the set of linear correlation matrices of radially symmetric Bernoullirandom vectors. This insight also appears in Theorem 1 of Huber and Mari´c (2019).We now turn to the problem of determining the attainable higher order concordance probabilities whenlower order probabilities are fixed. The following example illustrates what happens in four dimensions whenall pairwise Kendall rank correlations are equal and form a so-called equicorrelation matrix.
Example 1.
Let C be a copula in dimension d = 4 with Kendall rank correlation matrix P τ ( C ) equalto the equicorrelation matrix with off-diagonal element κ − ; in other words the bivariate concordanceprobabilities κ { i,j } for all pairs of random variables are equal to κ . The even concordance signature is avector of the form κ = (1 , κ, κ, κ, κ, κ, κ, κ ) where κ = κ { , , , } . It is straightforward to verify that thelinear system given in (18) is solved by the weight vector w = ( κ , w ( κ, κ ) , w ( κ, κ ) , w ( κ, κ ) , w ( κ, κ ) , w ( κ, κ ) , w ( κ, κ ) , w ( κ, κ )) , w ( κ, κ ) = (3 κ − / − κ and w ( κ, κ ) = 1 − κ + κ . Since the weights must satisfy (cid:54) w i ( κ, κ ) (cid:54) , the concordance probabilities must satisfy κ ∈ [1 / , and κ ∈ [max(2 κ − , , (3 κ − / . The left panel of Figure 2 shows the set of attainable values for τ = 2 κ − and τ = (8 κ − / .It is notable that the mere fact that the bivariate concordance probabilities are equal limits the range ofattainable correlations significantly. In principle, for any copula, the pair ( τ, τ ) must always lie in therectangle [ − , × [ − / , shown in the plot. However, in the case of equicorrelation, the attainable set isconsiderably smaller.In general, attainability and compatibility problems can be solved by linear programming. Let S ⊂ E ( D ) be a label set strictly contained in the even power set and suppose we want to test whether the vector κ = ( κ I : I ∈ S ) with label set S is attainable or, in other words, whether it is the partial concordancesignature of some copula C . If it is attainable, we then want to determine the attainable values for themissing concordance probabilities that make up the even concordance signature.Let A (1) d be the | S | × d − -matrix consisting of the rows of A d that correspond to S ; let A (2) d be thematrix formed of the remaining rows of A d . Consider the set of weight vectors { w : A (1) d w = κ , w (cid:62) } and note that every element of the set satisfies the sum condition on the weights, since S is a label setcontaining ∅ and the first row of A (1) d consists of ones. If κ is an attainable partial concordance signaturethen this set of weight vectors is non-empty and forms a convex polytope, that is, a set of the form (cid:40) m (cid:88) i =1 µ i w i , m (cid:88) i =1 µ i = 1 , µ i (cid:62) , i = 1 , . . . , m (cid:41) . (21)In this case it is possible to use the method of Avis and Fukuda (1992) to find the vertices w , . . . , w m ofthe polytope. The set of attainable even concordance signatures containing the partial signature κ is thengiven by the convex hull of the points κ i = A d w i , i ∈ { , . . . , m } , while the set of attainable unspecifiedelements of the concordance signature is given by the convex hull of the points { κ (2) i = A (2) d w i , i = 1 , . . . , m } . (22)As the dimension increases and the discrepancy between the length of κ and the length of the evensignature increases, the vertex enumeration algorithm can become computationally infeasible. In such caseswe may be content to simply test κ for attainability by finding a single even concordance signature thatcontains κ . To do so, we can attempt to solve the optimization problem min (cid:107) A (2) d w (cid:107) : A (1) d w = κ , w (cid:62) (23)where (cid:107) · (cid:107) denotes the Euclidean norm. This is a standard minimization problem with both equality andinequality constraints. The putative partial concordance signature κ is attainable if a solution exists and thesolution to the optimization problem will be the weight vector which gives the (collectively) smallest valuesfor the missing concordance probabilities. To get the (collectively) largest values we could solve min (cid:107) A (2) d w − (cid:107) : A (1) d w = κ , w (cid:62) . (24)Note also that if we are only concerned with finding the smallest or largest missing values for certain missingconcordance probabilities, then we can drop rows of A (2) d . Example 2.
For d = 5 let a partial concordance signature be given by κ { i,j } = 2 / for all pairs of randomvariables and κ { , , , } = κ { , , , } = 0 . . To complete the concordance signature, 3 further concordanceprobabilities are required: κ { , , , } , κ { , , , } and κ { , , , } . Using the method of Avis and Fukuda (1992)the set of possible weight vectors is non-empty and has nine vertices; thus the specified partial signature isattainable. The set of attainable values for the missing values forms a polytope in three dimensions which isshown in Figure 2. 14 . . . . . . tau−2D t au − D Figure 2: Left panel: set of attainable values of ( τ, τ ) in Example 1. Right panel: Convex polytope ofattainable fourth order concordance probabilities in Example 2. The concordance signature is identical for the copulas of all elliptical distributions with the same correlationmatrix P . This follows because the individual probabilities of concordance are identical for all such copulas.This is proved in Genest et al. (2011, Section 2.1) in the context of an analysis of multivariate Kendall’s taucoefficients.Let X have an elliptical distribution centred at the origin with dispersion matrix equal to the correlationmatrix P and assume that P ( X = ) = 0 . If X ∗ is an independent copy of X , then the concordanceprobabilities (1) are given by κ I = 2 P ( X I < X ∗ I ) = 2 P ( X I − X ∗ I < ) . The random vector X I − X ∗ I also has an elliptical distribution centred at the origin. Using a standard stochastic representation forelliptical distributions (see Fang et al. (1990) or McNeil et al. (2015)) we can write X I − X ∗ I d = R A S and X d = R A S where S is a random vector uniformly distributed on the unit sphere, A is a matrix such that AA (cid:62) = P and R and R are positive scalar random variables that are both independent of S . It followsthat κ I = 2 P ( X I − X ∗ I < ) = 2 P (cid:16) ( R A S ) I < (cid:17) = 2 P (cid:16) ( R A S ) I < (cid:17) = 2 P ( X I < ) . (25)From this calculation, we see that the positive scalar random variable R plays no role, so that theconcordance probabilities κ I are the same for any elliptical random vector with the same dispersion matrix P = AA (cid:62) . Moreover, they are equal to twice the orthant probabilities for a centred elliptical distributionwith dispersion matrix P . In practice it is easiest to calculate the orthant probabilities of a multivariatenormal distribution X ∼ N d ( , P ) and this is the approach we take in our examples. The results for the15 × correlation matrix P = 116
16 1 2 3 4 51 16 6 7 8 92 6 16 10 11 123 7 10 16 13 144 8 11 13 16 155 9 12 14 15 16 (26)are given in Table 1.Every linear correlation matrix can be the correlation matrix of a multivariate elliptical (or multivariatenormal) distribution. We now show by means of a counterexample that an analogous statement is not trueof Kendall rank correlation matrices.
Example 3.
The positive-definite correlation matrix P τ = − . − .
29 0 . − .
19 1 − .
34 0 . − . − .
34 1 − . .
49 0 . − .
79 1 (27)is a Kendall rank correlation matrix but is not the Kendall rank correlation matrix of an elliptical distribution.The elements of this matrix are attainable values for the Kendall’s tau coefficients τ { i,j } if the correspondingconcordance probabilities κ { i,j } = (1 + τ { i,j } ) / are attainable. Using the methods of the previous sectionwe can verify that κ = ( κ ∅ , κ { , } , κ { , } , κ { , } , κ { , } , κ { , } , κ { , } ) is an attainable partial concordancesignature. Solving the linear programming problem (23) gives the weight vector w = (0 . , . , . , , . , . , . , . corresponding to the minimum attainable fourth order concordance probability of 0.04. Solving the linearprogramming problem (24) gives the weight vector w = (0 . , . , . , . , . , . , . , corresponding to the maximum attainable fourth order concordance probability of 0.0425. In this case w and w are precisely the two vertices of the polytope of attainable weights given by the set (21), which takesthe form of a line segment connecting w and w . Any weight vector in this set will give the Kendall rankcorrelation matrix P τ .Let us assume that P τ in (27) corresponds to an elliptical copula. Lindskog et al. (2003) and Fang andFang (2002) have shown that the Kendall rank correlation matrix of an elliptical copula with correlationmatrix P is given by the componentwise transformation P τ = 2 π − arcsin( P ) . It must be the case that P = sin( πP τ / is the correlation matrix of the elliptical copula. However, by calculating the eigenvalueswe find that P is not positive semi-definite, which is a contradiction.We now turn to the copula of the multivariate Student t distribution C tν,P with degree of freedom pa-rameter ν and correlation matrix parameter P . It is unusual to consider this copula for degrees of freedom ν < , but we consider the limiting behaviour as ν → . The practical implication of the next result isthat C tν,P with ν very small provides an absolutely continuous parametric model that can approximate themixture of extremal copulas that shares its concordance signature with all elliptical distributions with corre-lation matrix P . The proof relies on some limiting results for the univariate and multivariate t distributionwhich are collected in the Appendix. 16able 1: Results for the correlation matrix P in (26). k S k w k I κ I { , , , , , } ∅ { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , } { , , , , , } { , , , , , } heorem 3. As ν → the d -dimensional t copula C tν,P converges pointwise to the unique extremal mixturecopula that shares its concordance signature.Proof. Let the function h ν ( w, s ) be defined by h ν ( w, s ) = F ν (cid:16) G − d,ν ( w ) A s (cid:17) , w ∈ (0 , , s = ( s , . . . , s d ) , s (cid:62) s = 1 , where F ν is the cdf of a t distribution with ν degrees of freedom, G d,ν is the cdf of the radial componentof a d -dimensional multivariate t distribution with ν degrees of freedom and A is a d × d matrix such that AA (cid:62) = P ; such a matrix can be constructed for any positive semi-definite P . Let S = ( S , . . . , S d ) be uniformly distributed on the unit sphere and let W be an independent uniform random variable. Then X = G − d,ν ( W ) A S has a multivariate t distribution and U = h ν ( W, S ) has joint cdf C tν.P . We want to showthat the joint cdf of h ν ( W, S ) converges to the joint cdf of an extremal mixture as ν → .We first argue that the random vector given by A S satisfies P (cid:0) ( A S ) j = 0 (cid:1) = 0 . Let a j denote the j throw of A . If a j = then the j th row and column of P would consist of zeros implying that the j th marginof the multivariate t distribution of X was degenerate; this case can be discounted because the t copula withsuch a matrix P is not defined. Suppose therefore that P ( a (cid:62) j S = 0) > for a j (cid:54) = . If R is the radialrandom variable corresponding to multivariate normal, then P ( a (cid:62) j R S = 0) = P ( a (cid:62) j Z = 0) > , where Z is a vector of d independent standard normal variables. However a (cid:62) j Z is univariate normal with variance a (cid:62) j a j > and cannot have an atom of mass at zero.We can therefore define the set A = { s : s (cid:62) s = 1 , ( A s ) j (cid:54) = 0 , j = 1 , . . . , d } such that P ( S ∈ A ) = 1 .Given that S = s ∈ A , then { U j (cid:54) u j } = (cid:26) W (cid:54) G d,ν (cid:18) F − ν ( u j )( A s ) j (cid:19)(cid:27) if ( A s ) > , { U j (cid:54) u j } = (cid:26) W (cid:62) G d,ν (cid:18) F − ν ( u j )( A s ) j (cid:19)(cid:27) if ( A s ) < ,and hence the conditional distribution function of U given S = s has the form P ( U (cid:54) u , . . . , U d (cid:54) u d | S = s )= P (cid:18) max j (cid:54)∈ I A s (cid:26) G d,ν (cid:18) F − ν ( u j )( A s ) j (cid:19)(cid:27) (cid:54) W (cid:54) min j ∈ I A s (cid:26) G d,ν (cid:18) F − ν ( u j )( A s ) j (cid:19)(cid:27)(cid:19) = (cid:18) min j ∈ I A s (cid:26) G d,ν (cid:18) F − ν ( u j )( A s ) j (cid:19)(cid:27) − max j (cid:54)∈ I A s (cid:26) G d,ν (cid:18) F − ν ( u j )( A s ) j (cid:19)(cid:27)(cid:19) + , where I A s is the set of indices j for which ( A s ) j > . Writing, for any u ∈ (0 , d , C tν.P ( u ) = P ( U (cid:54) u , . . . , U d (cid:54) u d ) = (cid:90) P ( U (cid:54) u , . . . , U d (cid:54) u d | S = s ) dF S ( s ) , we can use Proposition 7 in the Appendix and Lebesgue’s Dominated Convergence Theorem to concludethat C tν.P ( u ) converges, as ν → , to C ( u ) = (cid:90) (cid:18) min j ∈ I A s (2 u j − + − max j (cid:54)∈ I A s (1 − u j ) + (cid:19) + dF S ( s ) . (28)We now show that this limit is a mixture of extremal copulas. To this end, consider the random vector (cid:0) sign( A S ) + (cid:1) / . This has the same distribution as the multivariate Bernoulli random vector Y whose18istribution is defined by the probabilities p I = P ( Y I = ) = P (cid:0) ( A S ) j > , j ∈ I (cid:1) for I ⊆ D ; the randomvectors (cid:0) sign( A S ) + (cid:1) / and Y differ only on the null set where components of A S are zero. Moreover,the distribution of Y is radially symmetric since the spherical symmetry of S implies A S d = − A S whichin turn implies Y d = − Y . The limiting distribution (28) may be written in the form C ( u ) = (cid:88) y ∈{ , } d (cid:18) min j : y j =1 (2 u j − + − max j : y j =0 (1 − u j ) + (cid:19) + P ( Y = y ) and using the index set notation defined in Section 3 this may also be written as C ( u ) = d − (cid:88) k =1 (cid:32) min j ∈ J k (2 u j − + − max j ∈ J (cid:123) k (1 − u j ) + (cid:33) + P ( Y = s k )+ d − (cid:88) k =1 (cid:32) min j ∈ J (cid:123) k (2 u j − + − max j ∈ J k (1 − u j ) + (cid:33) + P ( Y = − s k ) . Setting P ( Y = s k ) = P ( Y = − s k ) = 0 . w k as in Section 3.2 we obtain C ( u ) = (cid:80) d − k =1 w k C k ( u ) ,where C k ( u ) = (cid:32) min j ∈ J k (2 u j − + − max j ∈ J (cid:123) k (1 − u j ) + (cid:33) + + (cid:32) min j ∈ J (cid:123) k (2 u j − + − max j ∈ J k (1 − u j ) + (cid:33) + and we need to check that C k is in fact the k th extremal copula C ( k ) given by (5). To do so, we have todistinguish the four cases below:(i) Suppose that there is at least one j ∈ J k and at least one j ∈ J (cid:123) k such that u j (cid:54) . . Then C k ( u ) =0 = C ( k ) ( u ) .(ii) Suppose that for all j ∈ J k , u j > . and there exists at least one j ∈ J (cid:123) k such that u j (cid:54) . . Then C k ( u ) = (cid:32) min j ∈ J k (2 u j − − max j ∈ J (cid:123) k (1 − u j ) (cid:33) + = 2 (cid:32) min j ∈ J k u j + min j ∈ J (cid:123) k u j − (cid:33) + = 2 C ( k ) ( u ) . (iii) The case when for all j ∈ J (cid:123) k , u j > . and there exists at least one j ∈ J k such that u j (cid:54) . isanalogous to case (ii) and is omitted.(iv) Suppose that for all j ∈ D , u j > . . Then C k ( u ) = min j ∈ J k (2 u j −
1) + min j ∈ J (cid:123) k (2 u j −
1) = 2 (cid:32) min j ∈ J k u j + min j ∈ J (cid:123) k u j − (cid:33) = 2 C ( k ) ( u ) . Finally, we need to verify that the concordance signature of the limiting mixture of extremal copulas is thesame as the concordance signature of C tν,P for any ν > . If κ I denotes a concordance probability for the t copula, we need to show that κ I = (cid:80) d − k =1 w k a I,k , which is the corresponding concordance probability forthe limit. Recall that the vector X = G − d,ν ( W ) A S has a multivariate t distribution. Equation (25) impliesthat κ I = 2 P (cid:16) G − d,ν ( W )( A S ) I < (cid:17) = 2 P (cid:16) ( A S ) I < (cid:17) = 2 P ( Y I = ) = 2 P ( Y I = ) = d − (cid:88) k =1 w k a I,k , where the final step uses the reasoning employed in the proof of Theorem 1.19 Figure 3: Scatterplot of data with distribution C tν,P when ν = 0 . and P is the × matrix with elements ρ = 0 . , ρ = 0 . and ρ = 0 . .Table 2: Probabilities associated with diagonals of unit cube for copula in Figure 3.Diagonal Probability (0 , , ↔ (1 , , (0 , , ↔ (1 , , (0 , , ↔ (1 , , (0 , , ↔ (1 , , C tν,P when ν = 0 . and P = ( ρ ij ) is the × matrix withelements ρ = 0 . , ρ = 0 . and ρ = 0 . . Clearly the points are distributed very close to the fourdiagonals of the unit cube. The limiting weights attached to extremal copulas associated with each diagonalare given in Table 2. Remark 3.
The proof holds even when the matrix P is not of full rank. However, because in such cases thecopula is distributed on a strict subspace of the unit hypercube [0 , d , the limiting extremal mixture copulahas zero mass on certain diagonals of the hypercube. Suppose, for example, that rows i and j of the matrix A satisfying AA (cid:62) = P are identical. Then the components Y i and Y j of the vector Y defined in the proofare identical, i.e. they are both or both . For any vector s k such that s k,i (cid:54) = s k,j it must be the case that w k = P ( Y = s k ) + P ( Y = − s k ) = 0 and so the k th diagonal would have zero mass. Consider a random sample X , . . . , X n from an unknown distribution with copula C and continuousmarginals F , . . . , F d . In this section, we explain how the full concordance signature ˜ κ ( C ) can be estimatedintrinsically, i.e. in such a way that the estimated signature is attainable. We do this under the assumption20hat there are no ties in the data; this is not restrictive because the continuity of the margins ensures theabsence of ties with probability .For any index set I with | I | (cid:62) , empirical estimators of κ I = κ ( C I ) can be derived from empiricalestimators of τ I = τ ( C I ) using (3). When d = 2 and I = { k, (cid:96) } for some k (cid:54) = (cid:96) ∈ { , . . . , d } , the classicalestimator of τ I going back to Kendall (1938) and Hoeffding (1947) is τ I,n = − n ( n − (cid:88) i (cid:54) = j I { X ik (cid:54) X jk ,X i(cid:96) (cid:54) X j(cid:96) } . This is a special case of the estimator of τ I for | I | (cid:62) proposed and investigated by Genest et al. (2011),which is given by τ I,n = 12 | I |− − (cid:110) − | I | n ( n − (cid:88) i (cid:54) = j (cid:89) k ∈ I I { X ik (cid:54) X jk } (cid:111) . Plugging this estimator into (3) yields an empirical estimator of κ I of the form κ I,n = 2 n ( n − (cid:88) i (cid:54) = j (cid:89) k ∈ I I { X ik (cid:54) X jk } . From the theory of U-statistics (Hoeffding, 1948), we know that the empirical concordance signature ˜ κ n =( κ I,n : I ∈ P ( D )) satisfies √ n (˜ κ n − ˜ κ ) (cid:32) N ( , Σ) as n → ∞ , where ˜ κ = ˜ κ ( C ) , Σ is the variance-covariance matrix of the random vector with components C I ( U I ) + ¯ C I ( U I ) and ¯ C I is the survival function of C I . The following result shows that the empiricalconcordance signature ˜ κ n is in fact the concordance signature of a d -dimensional copula. Theorem 4.
Assuming that n (cid:62) and there are no ties in the sample, there exists a d -dimensional copula C n such that ˜ κ n = ˜ κ ( C n ) .Proof. Let Y ij = (cid:0) sign( X i − X j ) + 1 (cid:1) / for i (cid:54) = j and set (cid:98) w k = 2 n ( n − (cid:88) i 21s the margin of C n corresponding to the index set I . This can be seen as follows κ ( C n,I ) = d − (cid:88) k =1 (cid:98) w k a I,k = d − (cid:88) k =1 n ( n − (cid:88) i Remark 4. While the probability of ties in a sample from a distribution with continuous margins is zero,rounding effects may lead to occasional ties in practice. In this case it is possible that some of the vectors Y ij have components equal to 0.5. Let us suppose that Y ij has k such values for k ∈ { , . . . , d } . A possibleapproach to incorporating this information in the estimator is to replace Y ij by the k vectors that have zerosand one in the same positions as Y ij , each weighted by − k , and to generalize (29) to be a weighted sum ofindicators. For example, the observation Y ij = (1 , , . , . would be replaced by (1 , , , , (1 , , , , (1 , , , and (1 , , , , each weighted by 1/4. This would still deliver estimates (cid:98) w k that are positive andsum to one and thus yield a proper concordance signature.We conclude with an example using real data which illustrates the use of the signature estimation methodand also indicates how the methods of this paper can be used when systems of rank correlations are incom-plete and missing values must be inferred from existing values. Example 4. We take the multivaritate time series of cryptocurrency prices (in US dollars) for Bitcoin,Ethereum, Litecoin and Ripple. From these data we compute the daily log-returns for the calendar year2017, giving us 365 4-dimensional data points. The estimated even concordance signature is κ n = (1 , . , . , . , . , . , . , . while the weight vector describing the extremal mixture copula C satisfying κ ( C ) = κ n is w = (0 . , . , . , . , . , . , . , . where all figures are given to 3 decimal places. The Kendall rank correlations of the copula C will exactlymatch the estimated values from the data.Suppose we did not have information about Ripple and instead had to defer to an expert for estimatesof the correlations with the other three cryptocurrencies. Using the methods of this paper, in particular bycomputing the convex hull of the set of points in (22), we obtain the polytope shown in the left panel ofFigure 4. The values specified by the expert must yield a point within this polytope or they are inconsistentwith the available information.Suppose that only the rank correlation τ { , } is known and taken to be the value implied by the evenconcordance signature above. Then the remaining two values must lie in the convex set shown in the rightpanel of Figure 4, which is a section of the 3-dimensional set, shown as a cut in the left panel. This iscertainly the case for the true estimated values which are shown as a point within the set.22 −1.0 −0.5 0.0 0.5 1.0 − . − . . . . {2,4} { , } Figure 4: Left panel: set of attainable Kendall rank correlations for Ripple paired with three other crytocur-rencies in Example 4. Right panel: set of attainable Kendall’s tau values for (Ripple, Etherium) and (Ripple,Litecoin) when the (Ripple, Bitcoin) value is known. Software The methods and examples in this paper are documented as vignettes in the R package KendallSignature at https://github.com/ajmcneil/KendallSignature . A Additional material on extremal mixture copulas Example 5. Let the joint distribution of ( U, Y , Y , Y ) be specified by P ( U (cid:54) u, Y = y , Y = y , Y = y ) = 18 (cid:18) u + ( − ( y + y + y ) θu (1 − u )4 (cid:19) for u ∈ [0 , and y i ∈ { , } , i ∈ { , , } . Note that this is an increasing function in u for any fixed ( y , y , y ) and defines a valid distribution. It may be easily verified, by summing over the outcomes forthe Y i variables, that the marginal distribution of U is standard uniform, while letting u → shows thatthe random vector Y = ( Y , Y , Y ) consists of iid Bernoulli variables with success probability . (and isradially symmetric). It may also be verified that the marginal distributions P ( U (cid:54) u, Y i = y i ) = 0 . u sothat the pairs ( U, Y i ) are independent for all i , while the marginal distributions P ( U (cid:54) u, Y i = y i , Y j = y j ) = 0 . u so that ( U, Y i , Y j ) are mutually independent for all i (cid:54) = j .Now consider the vector U = U Y + (1 − U )( − Y ) . Since the pairs ( U, Y i ) are independent it is easyto see that the components U i = U Y i + (1 − U )(1 − Y i ) are uniform, implying that the distribution of U is a copula. Since the triples ( U, Y i , Y j ) are mutually independent, the bivariate margins of U are extremalmixtures by Proposition 1. To calculate the copula C of U we observe that, for u = ( u , u , u ) , C ( u , u , u ) = (cid:88) i =1 (cid:16) P ( U (cid:54) u , Y = s k ) + P ( U (cid:54) u , Y = − s k ) (cid:17) = 2 (cid:88) i =1 P ( U (cid:54) u , Y = s k ) . The final equality follows because our model has the property that ( U, Y , Y , Y ) d = (1 − U, − Y , − Y , − Y ) . This implies that ( U , Y ) = ( U Y + (1 − U )( − Y ) , Y ) d = ( U , − Y ) . The copula C has 423istinct terms, each associated with a diagonal of the unit cube: C ( u , u , u ) = min( u , u , u ) + θ u , u , u ) max(1 − u , − u , − u )+ I { min( u ,u )+ u − (cid:62) } (cid:16) min( u , u ) − θ u , u ) max(1 − u , − u ) − u + θ u (1 − u ) (cid:17) + I { min( u ,u )+ u − (cid:62) } (cid:16) min( u , u ) − θ u , u ) max(1 − u , − u ) − u + θ u (1 − u ) (cid:17) + I { min( u ,u )+ u − (cid:62) } (cid:16) min( u , u ) − θ u , u ) max(1 − u , − u ) − u + θ u (1 − u ) (cid:17) . However, unless θ = 0 , C is not of the form (5) and is not an extremal mixture copula. Proof of Proposition 3. If U follows an extremal mixture copula then, by Proposition 1, it has thestochastic representation U d = U B + (1 − U )( − B ) for some Bernoulli random vector B and it is clearthat all the bivariate margins have the same structure. From the independence of U and B we obtain P (cid:16) U (cid:54) u | I { U j = U } , j (cid:54) = 1 (cid:17) = P (cid:0) U (cid:54) u | I { B = B } , . . . , I { B d = B } (cid:1) = P ( U (cid:54) u ) = u. Conversely, let us suppose that all the bivariate margins of U are mixtures of extremal copulas. This impliesthat almost surely, U takes values in the set (cid:92) i (cid:54) = j { u ∈ [0 , d : u j = u i or u j = 1 − u i } , which simplifies to the union E of the d − main diagonals of the unit hypercube, viz. E = d − (cid:91) k =1 { u ∈ [0 , d : u j = u (1 − s k,j )1 (1 − u ) s k,j , j (cid:54) = 1 } . Let E k = { U j = U (1 − s k,j )1 (1 − U ) s k,j , j (cid:54) = 1 } represent the event that U lies on the k th diagonal and set w k = P ( E k ) . By the law of total probability, we get C ( u ) = d − (cid:88) k =1 w k P ( U (cid:54) u , . . . , U d (cid:54) u d | E k ) = d − (cid:88) k =1 w k P ( U ∈ [max j ∈ J (cid:123) k (1 − u j ) , min j ∈ J k u j ] | E k ) . On the diagonals of the hypercube, { U ∈ E} ∩ { I { U j = U } = 1 − s k,j , j (cid:54) = 1 } = E k , so that w k = P { I { U j = U } = 1 − s k,j , j (cid:54) = 1 } and conditioning on the event E k is identical to conditioning on the event { I { U j = U } = 1 − s k,j , j (cid:54) = 1 } . We thus obtain that C ( u ) = d − (cid:88) k =1 w k P ( U ∈ [max j ∈ J (cid:123) k u j , min j ∈ J k u j ] | I { U j = U } = 1 − s k,j , j (cid:54) = 1) = d − (cid:88) k =1 w k C ( k ) ( u ) where the last equality follows from (11) and (5). This shows that C is indeed a mixture of extremal copulasas claimed. 24 Some limiting properties of the univariate and multivariate Student distribution as ν → Univariate Student t distribution Let F ν , F − ν and f ν denote the cdf, inverse cdf and density of a univariate t distribution with ν degrees offreedom. The cdf satisfies F ν ( x ) − . x Γ( ν +12 ) √ πν Γ( ν ) F (cid:18) , ν + 12 ; 32 ; − x ν (cid:19) , x ∈ R , ν > , (30)where F denotes the hypergeometric function and Γ the gamma function. We will show that, for fixed u (cid:54) = 0 . , the quantile function F − ν ( u ) is unbounded as a function of ν as ν → . To that end we first provethe following lemma. Lemma 3. lim ν → F ν ( x ) = 0 . for all x ∈ R .Proof. The lemma is trivially true for x = 0 so we consider x (cid:54) = 0 . Making the substitution y = x/ √ ν in (30) gives F ν ( √ νy ) − ν = 2 y Γ( ν +12 ) √ πν Γ( ν ) F (cid:18) , ν + 12 ; 32 ; − y (cid:19) and we can use the limits (Abramowitz and Stegun, 1965) lim ν → y F (cid:18) , ν + 12 ; 32 ; − y (cid:19) = y F (cid:18) , 12 ; 32 ; − y (cid:19) = ln (cid:16) y + (cid:112) y (cid:17) lim ν → ν +12 ) √ πν Γ( ν ) = lim ν → Γ( ν +12 ) √ π ν Γ( ν ) = lim ν → Γ( ν +12 ) √ π ν +22 ) = 1 to conclude that lim ν → F ν ( √ νy ) − ν = ln (cid:16) y + (cid:112) y (cid:17) = sign( y ) ln (cid:16) | y | + (cid:112) y (cid:17) . Reversing the earlier substitution and setting x = √ νy now gives lim ν → F ν ( x ) − ν sign( x ) (cid:16) ln (cid:0) | x | + √ ν + x (cid:1) − . ν (cid:17) = 1 . The result follows from the fact that the denominator tends to 0 as ν → . Lemma 4. lim ν → F − ν ( u ) = −∞ if u < . , if u = 0 . , ∞ if u > . .Proof. The case u = 0 is obvious, since F − ν (0 . 5) = 0 for all ν > . To show that lim ν → F − ν ( u ) = −∞ for u < . , we need to show that, for all k < , there exists δ such that ν < δ ⇒ F − ν ( u ) (cid:54) k . Supposewe fix an arbitrary k < . Since F ν ( k ) → . as ν → , there exists δ > such that ν < δ implies that F ν ( k ) > u , since u < . . But then, for any ν < δ , it follows that k ∈ { x : F ν ( x ) (cid:62) u } and hence F − ν ( u ) = inf { x : F ν ( x ) (cid:62) u } (cid:54) k .Analogously, to show that lim ν → F − ν ( u ) = ∞ for u > . , we need to show that, for all k > , thereexists δ such that ν < δ ⇒ F − ν ( u ) (cid:62) k . Suppose we fix an arbitrary k > . Since F ν ( k ) → . as ν → ,there exists δ > such that ν < δ implies that F ν ( k ) < u , since u > . . But then, for any ν < δ , itfollows that k / ∈ { x : F ν ( x ) (cid:62) u } . Since F ν ( x ) is a non-decreasing function of x , any y < k also satisfies y / ∈ { x : F ν ( x ) (cid:62) u } . Hence F − ν ( u ) = inf { x : F ν ( x ) (cid:62) u } (cid:62) k .25 ultivariate Student t distribution If the random vector X has a d -dimensional multivariate t distribution with ν degrees of freedom, then it hasthe stochastic representation X d = µ + RA S where S is a uniform random vector on the d -dimensional unitsphere, R is an independent, positive, scalar random variable such that R /d ∼ F ( d, ν ) (a Fisher–Snedecor F distribution), µ is a location vector and A is a matrix; see Section 6.3 in McNeil et al. (2015). Let G d,ν denote the cdf of the radial random variable R and g d,ν the corresponding density; the latter is given by g d,ν ( r ) = 2 r (cid:16) νr (cid:17) − d (cid:18) r ν (cid:19) − ν Γ (cid:0) ν + d (cid:1) Γ (cid:0) d (cid:1) Γ (cid:0) ν (cid:1) . (31)The following limiting result is key to our analysis of the limiting behaviour of the t copula as ν → .Note that the limiting function on the right-hand side is either the cdf of a random variable that is uniformlydistributed on [0 . , or the survival function of a random variable that is uniformly distributed on [0 , . ,depending on the sign of λ . Proposition 7. For a constant λ (cid:54) = 0 and any u ∈ [0 , , lim ν → G d,ν (cid:18) F − ν ( u ) λ (cid:19) = (cid:40) (2 u − + if λ > , (1 − u ) + if λ < . (32) Proof. When λ > , F d,ν ( u ) = G d,ν (cid:0) λ − F − ν ( u ) (cid:1) is a distribution function supported on [0 . , . Simi-larly, when λ < , F d,ν ( u ) = 1 − G d,ν (cid:0) λ − F − ν ( u ) (cid:1) is a distribution function supported on [0 , . . Thedensity of F d,ν is given by f d,ν ( u ) = g d,ν (cid:18) | F − ν ( u ) || λ | (cid:19) | λ | f ν (cid:0) F − ν ( u ) (cid:1) where u ∈ [0 . , when λ > and u ∈ (0 , . when λ < .Note that, when u = 0 . , F − ν ( u ) = 0 for all ν and (32) clearly holds; we thus restrict our analysis ofthe density to the case where u (cid:54) = 0 . . Using the notation x ν,u = F − ν ( u ) and the expression (31) we havethat f d,ν ( u ) = g d,ν (cid:18) | x ν,u || λ | (cid:19) | λ | f ν (cid:0) x ν,u (cid:1) = 2 | x ν,u | (cid:18) λ νx ν,u (cid:19) − d (cid:32) x ν,u λ ν (cid:33) − ν Γ (cid:0) ν + d (cid:1) Γ (cid:0) d (cid:1) Γ (cid:0) ν (cid:1) √ νπ Γ (cid:0) ν (cid:1) Γ (cid:0) ν +12 (cid:1) (cid:32) x ν,u ν (cid:33) ν +12 = 2 (cid:113) ν + x ν,u | x ν,u | (cid:18) λ νx ν,u (cid:19) − d (cid:32) x ν,u λ ν (cid:33) − ν (cid:32) x ν,u ν (cid:33) ν Γ (cid:0) ν + d (cid:1) Γ (cid:0) d (cid:1) √ π Γ (cid:0) ν +12 (cid:1) = 2 (cid:113) ν + x ν,u | x ν,u | (cid:124) (cid:123)(cid:122) (cid:125) (cid:18) λ νx ν,u (cid:19) − d (cid:124) (cid:123)(cid:122) (cid:125) (cid:32) λ (cid:0) ν + x ν,u (cid:1) λ ν + x ν,u (cid:33) ν (cid:124) (cid:123)(cid:122) (cid:125) Γ (cid:0) ν + d (cid:1) Γ (cid:0) d (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) √ π Γ (cid:0) ν +12 (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) . 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