Concordance Measures for Multivariate Non continuous Random Vectors

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Concordance measures for multivariate non-continuous random vectors

Mhamed Mesfioui, Jean-François Quessy
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DOI:
10.1016/j.jmva.2010.06.011
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Journal of Multivariate Analysis 101 (2010) 2398–2410  Contents lists available at ScienceDirect  Journal of Multivariate Analysis journal homepage: www.elsevier.com/locate/jmva  Concordance measures for multivariate non-continuous random vectors Mhamed Mesfioui, Jean-François Quessy ∗ Département de mathématiques et d'informatique, Université du Québec à Trois-Rivières, C. P. 500, Trois-Rivières (Québec), Canada G9A 5H7  article  info  Article history: Received 15 May 2009 Available online 18 June 2010 AMS subject classifications: 62H20 62G99 Keywords: Discontinuous distributions Copula Kendall's tau Multivariate concordance Spearman's rho Spearman's footrule  abstract A notion of multivariate concordance suitable for non-continuous random variables is defined and many of its properties are established. This allows the definition of multivariate, non-continuous versions of Kendall's tau, Spearman's rho and Spearman's footrule, which are concordance measures. Since the maximum values of these association measures are not +1 in general, a special attention is given to the computation of upper bounds. The latter turn out to be multivariate generalizations of earlier findings made by Nešlehová (2007) [9] and Denuit and Lambert (2005) [2]. They are easy to compute and can be estimated from a data set of (possibly) discontinuous random vectors. Corrected versions are considered as well. © 2010 Elsevier Inc. All rights reserved.  1. Introduction In many applications, measuring the strength of the dependence between the components of a random vector is of a primary interest. To this end, multivariate versions of popular association measures like Kendall's tau, Spearman's rho and Spearman's footrule, to name only a few, have been proposed. These measures, and many other dependence indices, are based on concordance. For continuous distributions, the concordance between two random vectors is written as a functional of their underlying copulas. For non-continuous distributions, this is not possible since the copula is;  no longer unique. See [3] for a survey of the pros and cons of using copulas in the modelling of multivariate non-continuous observations. In the bivariate case, Mesfioui and Tajar [6] Denuit and Lambert [2] and Nes̆lehová [9] proposed to apply a transformation to the random vectors in order that the resulting distributions are continuous. This is equivalent in selecting one of the many possible copulas of a non-continuous joint distribution function. Then, usual concordance is computed with respect to that chosen copula. The main contribution of this paper consists in extending this idea to the multivariate case. In other words, the goal is to unify the works by Nes̆lehová [9], who deals with the bivariate discontinuous case, and those about multivariate extensions of association measures in the continuous case, e.g. [7]. To this end, a new definition of concordance that will generalize that in the continuous case will be considered. The latter takes into account all the possible ties that can occur with non-null probability. Moreover, it will be shown that the newly introduced concordance function can be written in terms of a linear operator applied to the usual concordance operator. As a result, concordance for non-continuous random vectors will inherit many of the properties of multivariate concordance for continuous vectors. The paper is organized as follows: in Section 2, a generalization of bivariate concordance to the multivariate case due to [7] is recalled. In Section 3, this definition is broaden in order to encompass the case of non-continuous random vectors, and  ∗  Corresponding author. E-mail address: Jean-Francois.Quessy@UQTR.CA (J.-F. Quessy).  0047-259X/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmva.2010.06.011  M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  2399  many properties are established. Then, in Section 4, three measures of concordance, namely Kendall's tau, Spearman's rho and Spearman's footrule, are generalized to the multivariate non-continuous case. Section 5 is devoted to the computation of explicit and easy to handle upper bounds, which suggests corrected versions for the association measures. The estimation from possibly non-continuous observations is considered in Section 6, along with illustrations on a real data set. Closing remarks are offered in Section 7. 2. Multivariate concordance: the continuous case Let (X11 , X12 ) and (X21 , X22 ) be random vectors with respective continuous cumulative distribution functions H1 and H2 . It is supposed that H1 and H2 have the same marginal distributions, i.e. H1 (x, ∞) = H2 (x, ∞) = F1 (x) and H1 (∞, x) = H2 (∞, x) = F2 (x). The probability of concordance associated with H1 and H2 is defined by  Q(H1 , H2 ) = P {(X11 − X21 )(X12 − X22 ) > 0} ,  (1)  while the probability of discordance is simply Q̄(H1 , H2 ) = P{(X11 − X21 )(X12 − X22 ) < 0}. Kendall's tau τ and Spearman's rho ρ for a given bivariate distribution H are defined in terms of differences between probabilities of concordance and discordance. Specifically, τ = Q(H , H ) − Q̄(H , H ) and ρ = 3{Q(H , Π ) − Q̄(H , Π )}, where Π (x1 , x2 ) = F1 (x1 )F2 (x2 ) is the bivariate distribution associated with independence. In dimension greater than two, it is however unclear how to define discordance. For that reason, multivariate extensions of measures of dependence are usually defined in terms of concordance only. Hence, in order to extend the notion of concordance to a multivariate setting, a possibility is to define d-variate concordance in terms of bivariate concordance. To this end, consider the set Γ (F1 , . . . , Fd ) of all d-variate distribution functions whose margins are F1 , . . . , Fd and let X1 , X2 be random vectors with respective distribution functions H1 , H2 ∈ Γ (F1 , . . . , Fd ). Nelsen [7] defined multivariate concordance as the probability of concordance among all d(d − 1)/2 possible pairs. Then one can show that the concordance between X1 and X2 is  Q(H1 , H2 ) = P(X1 < X2 ) + P(X2 < X1 ),  (2)  where here and in the sequel, inequalities are taken componentwise. Remark 1. In the case of continuous random variables, Q(H1 , H2 ) = Q(C1 , C2 ), where C1 , C2 are the unique copulas associated with H1 , H2 through Ck (u) = Hk F1 (u1 ), . . . , Fd (ud ) ,    −1  −1  k = 1, 2,  where u = (u1 , . . . , ud ). Then, one can show easily that  Q(H1 , H2 ) =  Z [0,1]d  {C1 (u)dC2 (u) + C2 (u)dC1 (u)} .  This relationship between concordance and dependence functions (i.e. copulas) however fails to hold when one or more of the components of a random vector are discontinuous; since then, the copula is no longer unique. For more details on copulas, the reader is referred to the excellent monograph by Nelsen [8]. Many multivariate extensions of popular bivariate measures of association for a continuous random vector H are defined in terms of Q. In particular, d-variate versions of Kendall's tau, Spearman's rho and Spearman's footrule are written, respectively, as affine transformations of Q(C , C ), Q(C , Πd ) and Q(C , Md ), where C is the copula of H, Md (u) = min(u1 , . . . , ud ) is the Fréchet–Hoeffding upper bound and Πd (u) = u1 · · · ud . Versions of these measures that encompass the non-continuous case will be described later in details. They are based on an alternate definition of concordance, which is the subject of the next section. 3. Multivariate concordance: the non-continuous case 3.1. A generalization of concordance Remark 1 emphasizes on the fact that concordance of two continuous random vectors is uniquely described in terms of their copulas. In the non-continuous case, however, the form of the copula is only unique on ran F1 × · · · × ran Fd . Indeed, there are many possible copulas associated with a given non-continuous distribution function. Thus, if one wants a general definition of concordance based on copulas that encompass the non-continuous case, it is necessary to select one of these possible copulas. In the bivariate case, this approach has been successfully adopted by Nes̆lehová [9], where concordance is generalized to the concordance of suitably transformed random variables. A multivariate analog is described next. For that purpose, let H1 , H2 ∈ Γ (F1 , . . . , Fd ) and define  Ψ (x, u) = (ψ1 (x1 , u1 ), . . . , ψd (xd , ud )) ,  (3)  2400  M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  where ψ` (x, u) = F` (x− ) + u{F` (x) − F` (x− )}. For reasons that will become clear later, the space R × [0, 1] is endowed with the total lexicographical order relationship, i.e.  (x, u)  (x0 , u0 ) ⇔ (x < x0 ) or (x = x0 , u 6 u0 ). Now let U1 and U2 be independent and uniformly distributed on [0, 1]d . Suppose that U1 and X1 (respectively U2 and X2 ) are independent and defined on some common probability space. Note that the distributions of the random vectors X01 = Ψ (X1 , U1 ) and X02 = Ψ (X2 , U2 ) are continuous with standard uniform margins. This is a consequence of Lemma 3 of [9]. In other words, the distributions of X01 and X02 are copulas, denoted as C1 and C2 . Then, concordance between X1 and X2 is defined by Q̃(H1 , H2 ) = Q(C1 , C2 ). When F1 , . . . , Fd are continuous, ψ` (X` , U` ) = F` (X` ), ` ∈ Sd = {1, . . . , d}, are the usual probability integral transformations. Hence, the distribution of Ψ (X, U) is the unique copula C of X. As a consequence, Q̃(H1 , H2 ) = Q(H1 , H2 ) in the continuous case. Apart from being a proper generalization of concordance, Q̃ possesses many interesting properties, as will be seen later. Indeed, many properties of concordance in the continuous case also apply to Q̃. The practical aspects of this definition based on transformation (3) are, however, less clear. The next lemma provides a nice formula that relates Q̃ and Q. To this end, consider the random vectors X1 = (X11 , . . . , X1d ) and X2 = (X21 , . . . , X2d ) and introduce the operators T1 , . . . , Td satisfying T` P (X1 < X2 ) = P (X11 < X21 , . . . , X1` 6 X2` , . . . , X1d < X2d ) , where ` ∈ Sd = {1, . . . , d}. Now for arbitrary measurable sets A1 , . . . , Ad and B1 , . . . , Bd such that A` ∩ B` = ∅, note that  ! ( ) d \ \ \ X P A` ∩ B` P (A` ∪ B` ) = `=1  `∈A  A⊆Sd  (4)  `6∈A  and d Y  ! (a` + b` ) =  `=1  X Y A⊆Sd  ! Y  a`  `∈A  .  b`  (5)  `6∈A  These formulas are used repeatedly throughout the paper. Lemma 2. Let X1 ∼ H1 and X2 ∼ H2 be independent, where H1 , H2 ∈ Γ (F1 , . . . , Fd ). Let U1 (respectively U2 ) be uniform on [0, 1]d and independent of X1 (respectively X2 ). If U1 and U2 are independent, then  Q̃(H1 , H2 ) = T Q(H1 , H2 ),   d  Y T` + 1  where T =  2  `=1  .  (6)  Proof. First note that  Q̃(H1 , H2 ) = P {Ψ (X1 , U1 ) < Ψ (X2 , U2 )} + P {Ψ (X2 , U2 ) < Ψ (X1 , U1 )} . From the definition of , one sees that ψ` (X1` , U1` ) < ψ` (X2` , U2` ) is equivalent to (X1` < X2` ∪ X1` = X2` , U1` 6 U2` ), ` ∈ Sd . In view of formula (4), one has P {Ψ (X1 , U1 ) < Ψ (X2 , U2 )} = P  d \  ! ψ` (X1` , U1` ) < ψ` (X2` , U2` )  `=1  =P  d \  ! (X1` < X2` ∪ X1` = X2` , U1` 6 U2` )  `=1  ! =  X  \  P  `6∈A  A⊆Sd  =  X 1 A⊆Sd  X1`  \ < X2` , [X1` = X2` , U1` 6 U2` ]  2|A|  `∈A  ! P  \ `6∈A  X1` < X2` ,  \  X1` = X2`  `∈A  This last expression may be reduced to  X Y  T` − 1  2  A⊆Sd `∈A  P (X1 < X2 ) ,  using the fact that  ! (Ti − 1) P  \ `6∈A  X1` < X2` , X1i < X2i  ! =P  \ `6∈A  X1` < X2` , X1i = X2i  .  .  M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  2401  Now by using formula (5) with a` = (T` − 1)/2 and b` = 1, it follows that  X Y  T` − 1   P (X1 < X2 ) =  2  A⊆Sd `∈A   d  Y T` + 1 2  `=1  P (X1 < X2 )  = T P (X1 < X2 ) . Similarly, P {Ψ (X2 , U2 ) < Ψ (X1 , U1 )} = T P(X2 < X1 ). As a consequence, one has Q̃(H1 , H2 ) = T P (X1 < X2 ) + T P (X2 < X1 ) = T Q(H1 , H2 ).  Using formula (5) again, one observes that 1  Q̃(H1 , H2 ) =  (  2d  ) XY  T` Q(H1 , H2 ) ,  (7)  A⊆Sd `∈A  which can be viewed as the mean of the concordances account all the possible ties that can occur.  Q  `∈A T`  Q(H1 , H2 ), A ⊆ Sd . In other words, Q̃(H1 , H2 ) takes into  3.2. Properties of Q̃ It will be seen that the concordance function Q̃ has many enviable properties. As a starting point, the next lemma relates the distribution and survival functions of X0 = Ψ (X, U) to that of X, where the latter is assumed to be independent of U. To this end, proceed as in [9] and define, for s = (s1 , . . . , sd ) ∈ [0, 1]d , x(s) = (x1 (s1 ), . . . , xd (sd )) where for ` ∈ Sd , x` (s` ) = u` ( s ` ) =   1,   F`−1 (s+ `)  and u(s) = (u1 (s1 ), . . . , ud (sd )), and if F` x` (s` )− = F` {x` (s` )} ;    s` − F` x` (s` )−    F` {x` (s` )} − F` {x` (s` )− }  ,  otherwise.  Lemma 3. Let U be uniform on [0, 1]d and independent of X. Then, the distribution and survival functions of X0 = Ψ (X, U) are respectively d Y  C (s) =  ! {1 − u` (s` ) + u` (s` )T` } P {X < x(s)}  (8)  `=1  and d Y  C¯(s) =  ! {u` (s` ) + (1 − u` (s` ))T` } P {X > x(s)} .  (9)  `=1  Proof. Since for any ` ∈ Sd ,  {ω ∈ Ω |ψ` (X` (ω), U` (ω)) 6 s` } = {ω ∈ Ω | (X` (ω), U` (ω))  (x` (s` ), u` (s` ))} , one has from (4) and the definition of  that C (s) = P  d \  ! ψ` (X` , U` ) 6 s`  `=1  =P  d \  ! {X` < x` (s` ) ∪ [X` = x` (s` ), U` 6 u` (s` )]}  `=1  ! =  X Y A⊆Sd  `∈A  u` (s` ) P  ! \  X1` < X2` ,  `6∈A  \  X1` = X2`  `∈A  ! =  X Y A⊆Sd  =  d Y  `=1  u` (s` ) (T` − 1) P {X < x(s)}  `∈A  ! {1 − u` (s` ) + u` (s` )T` } P {X < x(s)}  2402  M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  where the last equality is obtained from (5) with al = u` (s` )(T` − 1) and bl = 1. A similar argument implies (9), which completes the proof.  In the purely discrete case, i.e. when ran F1 × · · · × ran Fd = Nd , [6,2], among others, suggested the transformation X0 = X + (U − 1) in the bivariate case. As noted by Nes̆lehová [9], the latter is a special case of Ψ . One can then show that x(s) = s and for any ` ∈ Sd , one has u` (s` ) = 1 − x` + bx` c, where bxc is the integer part of a real number x. Hence, the multivariate interpolation copula of a discrete random vector X is C (s) =  d Y  ! {x` − bx` c + (1 − x` + bx` c) T` } P (X < s) .  `=1  Now the expressions of C and C¯ in Lemma 3 are used to show that the concordance order between non-continuous random vectors is equivalent to the concordance order of their continuous versions. This gives another argument in favor of the continuous extension Ψ (X, U) as a suitable way to select one of the possible copulas of X. Recall that a vector X1 ∼ H1 ∈ Γ (F1 , . . . , Fd ) is said to be less concordant than X?1 ∼ H1 ∈ Γ (F1 , . . . , Fd ) if P(X1 ≤ x) 6 P(X?1 6 x) and P(X1 > x) ≤ P(X?1 > x) for all x in ran F1 × · · · × ran Fd . This will be denoted as X1 ≺Q X?1 . For more details on multivariate concordance, see [4]. Proposition 1. Let X01 = Ψ (X1 , U1 ) and X02 = Ψ (X2 , U2 ), where X1 and U1 (respectively X2 and U2 ) are independent and for k = 1, 2, Xk ∼ Hk ∈ Γ (F1 , . . . , Fd ). If U1 and U2 are independent and uniformly distributed on [0, 1]d , then X1 ≺Q X2 ⇔ X01 ≺Q X02 . Proof. (⇒) X1 ≺Q X2 means P(X1 < x) 6 P(X2 < x) and P(X1 > x) 6 P(X2 > x) for all x ∈ Rd . Then, from Lemma 3, one has for all s = (s1 , . . . , sd ) ∈ [0, 1]d that 0    P X1 6 s =  d Y  ! {1 − u` (s` ) + u` (s` )T` } P {X1 < x(s)}  `=1  6  d Y  !  {1 − u` (s` ) + u` (s` )T` } P {X2 < x(s)} = P X02 6 s ,  `=1  where the fact that `=1 {1 − u` (s` ) + u` (s` )T` } is a polynomial in T1 , . . . , Td with positive coefficients has been used.   Similarly, one has P X01 > s 6 P X02 > s .  Qd  (⇐) For all x in ran F1 × · · · × ran Fd , P(Xk ≤ x) = P X0k ≤ F(x) , k = 1, 2, where F(x) = (F1 (x1 ), . . . , Fd (xd )). Then,   P (X1 6 x) = P X01 6 F(x) 6 P X02 6 F(x) = P (X2 6 x). Analogously, one obtains P (X1 > x) 6 P (X2 > x).     As noted by Nelsen [7], Q(H1 , H2 ) is symmetric and increasing in its arguments. These properties are still valid for Q̃. Proposition 2. The following properties hold for Q̃:  (i) Q̃ is symmetric in its arguments, i.e. Q̃(H1 , H2 ) = Q̃(H2 , H1 ); (ii) Let X1 ∼ H1 , X2 ∼ H2 , X?1 ∼ H1? and X?2 ∼ H2? , where H1 , H2 , H1? , H2? ∈ Γ (F1 , . . . , Fd ). Then, X1 ≺Q X?1 and X2 ≺Q X?2 implies Q̃(H1 , H2 ) 6 Q̃(H1? , H2? ). Proof. Suppose that Xk and Uk (respectively X?k and U?k ) are independent and let Ck be the copula of Ψ (Xk , Uk ) and Ck? be the copula of Ψ (X?k , U?k ), k = 1, 2. For (i), Q̃(H1 , H2 ) = Q(C1 , C2 ) = Q(C2 , C1 ) = Q̃(H2 , H1 ). To establish (ii), note from Proposition 1 that Xk ≺Q X?k implies Ψ (Xk , Uk ) ≺Q Ψ (X?k , U?k ), k = 1, 2. Hence, Q̃(H1 , H2 ) = Q(C1 , C2 ) 6 Q(C1? , C2? ) = Q̃(H1? , H2? ).  4. Concordance measures for non-continuous random vectors Let H ∈ Γ (F1 , . . . , Fd ). The proposed population versions of Kendall's tau, Spearman's rho and Spearman's footrule for a possibly non-continuous random vector X ∼ H are respectively  τ̃d (H ) = ρ̃d (H ) = ϕ̃d (H ) =  2d−1 Q̃(H , H ) − 1 2d−1 − 1 d+1  ,  (10)  2d−1 Q̃(Πd , H ) − 1 ,  (11)  d+1  Q̃(H , Md ) − Q̃(Πd , Md ) , d−1  (12)  2d − d − 1    M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  2403  where Πd (x) = F1 (x1 ) · · · Fd (xd ) and Md (x) = min{F1 (x1 ), . . . , Fd (xd )}. Since Q̃ = Q in the continuous case, Eqs. (10)– (12) correspond to multivariate extensions already proposed in the literature. In particular, (10) and (11) are one of the possible extensions of Kendall's tau and Spearman's rho mentioned by Joe and Nelsen [4,7]. For continuous random vectors, the definition of ϕ̃d (H ) in the bivariate case dates back to [11]. Úbeda-Flores [13] proposed its multivariate extension. For a survey on measures of multivariate concordance, see [7]. Wolff [14] considered other types of measures of multivariate association, while [12] discusses the conditions that multivariate measures of association must satisfy. A practical justification of using τ̃d (H ), ρ̃d (H ) and ϕ̃d (H ) appears under independence. In fact, if H = Πd , a naive use of Kendall's tau, Spearman's rho and Spearman's footrule for continuous vectors would lead to measures that are not vanishing. This is not the case for their modified versions. Proposition 3. One has τ̃d (Πd ) = ρ̃d (Πd ) = ϕ̃d (Πd ) = 0. Proof. First compute T Q (Πd , Πd ) = 2   d  Y T` + 1 2  `=1  =  1 2d−1  d Y  P (X1` < X2` )  {P (X1` 6 X2` ) + P (X1` < X2` )} =  `=1  1 2d−1  ,  where the last equality uses the fact that P (X1` < X2` ) + P (X1` 6 X2` ) = P (X1` < X2` ) + P (X1` > X2` ) = 1. From (10) and (11), it is then clear that τ̃d (Πd ) = ρ̃d (Πd ) = 0. Showing that ϕ̃d (Πd ) = 0 is straightforward.    Remark 4. Proposition 3 can be seen as a consequence of the fact that the copula of Ψ (X, U) when X ∼ Πd is the independence copula CΠd (u) = u1 · · · ud . As an application of Proposition 2(ii), the measures of concordance defined in terms of Q̃ take larger values whenever the variables in a random vector are more associated in the sense of the concordance order. In particular, this is true for the modified versions of Kendall's tau, Spearman's rho and Spearman's footrule. Corollary 5. Let X1 ∼ H1 ∈ Γ (F1 , . . . , Fd ) and X?1 ∼ H1? ∈ Γ (F1 , . . . , Fd ). If X1 ≺Q X?1 , then τ̃d (H1 ) 6 τ̃d (H1? ), ρ̃d (H1 ) 6 ρ̃d (H1? ) and ϕ̃d (H1 ) 6 ϕ̃d (H1? ). In the continuous case, it has been noted by Úbeda-Flores [13] that tri-variate concordance measures can be expressed as the mean of the bivariate concordance measures of all 2-variate margins. The result can be generalized to measures of concordance based on Q̃. Proposition 4. Let H ∈ Γ (F1 , F2 , F3 ) and denote its bivariate margins by H 12 (x1 , x2 ) = H (x1 , x2 , ∞), H 13 (x1 , x3 ) = H (x1 , ∞, x3 ) and H 23 (x2 , x3 ) = H (∞, x2 , x3 ). Then  κ̃3 (H ) =     κ̃2 H 12 + κ̃2 H 13 + κ̃2 H 23 3  ,  where κ̃d is either τ̃d , ρ̃d or ϕ̃d . Proof. Consider H1 , H2 ∈ Γ (F1 , F2 , F3 ) together with their interpolation copulas C1 and C2 . From [13],  Q C112 , C212 + Q C113 , C213 + Q C123 , C223 − 1      Q̃(H1 , H2 ) = Q(C1 , C2 ) =  =    2    Q̃ H112 , H212 + Q̃ H113 , H213 + Q̃ H123 , H223 − 1 2  ,  where Ck`` and Hk`` , k ∈ {1, 2}, `, `0 ∈ {1, 2, 3}, are the bivariate margins of Ck and Hk , respectively. The conclusion follows from (10)–(12).  0  0  5. Upper bounds for the modified concordance measures Let X be an arbitrary random vector with distribution H ∈ Γ (F1 , . . . , Fd ). Then, it is always true that X ≺Q XMd , where X ∼ Md ∈ Γ (F1 , . . . , Fd ), with Md (x) = min{F1 (x1 ), . . . , Fd (xd )} being the Fréchet–Hoeffding upper bound. A particular case of Corollary 5 with H1? = Md yields τ̃d (H ) 6 τ̃d (Md ), ρ̃d (H ) 6 ρ̃d (Md ) and ϕ̃d (H ) 6 ϕ̃d (Md ). In general, these upper bounds cannot reach +1 as is the case under perfect association of continuous random variables. In the bivariate case, this Md  2404  M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  has been noted by Mesfioui and Tajar [6] and Denuit and Lambert [2], among others. The reason is that the copula of Ψ (X, U) when X ∼ Md do not coincide with the Fréchet–Hoeffding upper bound CMd (u) = min(u1 , . . . , ud ), unless all margins are continuous. A solution for having dependence measures whose values are +1 under Md is to use normalization. However, it turns out that τ̃d (Md ), ρ̃d (Md ) and ϕ̃d (Md ) depend on the marginal distributions in a complicated way. Hence, for practical reasons, bounds that are less sharp but easier to compute will be derived. As will be seen in the remaining of this section, they are simple functions of either D` = E{F` (X`− )} or some of the first d moments of ξ` = F` (X`− ) + F` (X` ) − 1, ` ∈ Sd . Although they are less sharp, it will be seen that these bounds are nevertheless attained in the case when the components of a random vector are linked together through strictly increasing functions. Hence, corrected versions arising from these bounds attain +1 under this particular situation of perfect association. This aspect is treated in Section 5.4. In the bivariate case, bounds for Kendall's tau and Spearman's rho have been derived. Namely, for any H ∈ Γ (F1 , F2 ), one has  p τ̃2 (H ) 6 2 min(D1 , D2 ) and ρ̃2 (H ) 6 3 var(ξ1 )var(ξ2 ). This bound for τ̃2 (H ) is due to [2], while that for ρ̃2 (H ) has been found by Nes̆lehová [9]. An application of Proposition 4 then yields upper bounds for Kendall's tau and Spearman's rho in the case d = 3. Namely, for any H ∈ Γ (F1 , F2 , F3 ),  τ̃3 (H ) 6  4  ρ̃3 (H ) 6  p  3  D(1) +  2 3  D(2)  and var(ξ1 )var(ξ2 ) +  p  var(ξ1 )var(ξ3 ) +  p  var(ξ2 )var(ξ3 ),  where D(1) 6 D(2) 6 D(3) . In the next two subsections, these bounds are generalized to the d-variate situation. A bound for Spearman's footrule is also obtained in Section 5.3. 5.1. A bound for Kendall's tau The following proposition describes an upper bound for τ̃d (H ). This is a multivariate analog of the bound proposed by Denuit and Lambert [2] in the bivariate case. Proposition 5. For all H ∈ Γ (F1 , . . . , Fd ),  τ̃d (H ) 6 τ̃max =  2d  d−1  ` X 1  2d−1 − 1 `=1  2  D(`) ,  where for ` ∈ Sd , D` = E{F` (X`− )} and D(1) 6 D(2) 6 · · · 6 D(d) . Proof. Let Y1 , Y2 ∼ Md ∈ Γ (F1 , . . . , Fd ) be independent. Note that Q(Md , Md ) = 2P(Y1 < Y2 ), so that Q̃(Md , Md ) = 2 T P(Y1 < Y2 ). In that case, one has the representations Y1 = (F1−1 (U ), . . . , Fd−1 (U )) and Y2 = (F1−1 (V ), . . . , Fd−1 (V )) in terms of independent standard uniform random variables U and V . For k ∈ {1, 2}, if YAk is the |A|-variate random vector whose components are Yk` , ` ∈ A, one can then write T P (Y1 < Y2 ) =  (  ) d Y T` + 1 2  `=1  = = =  1 2d 1 2d 1  X  Y  P (Y1 < Y2 )  T` P (Y1 < Y2 ) +  P(Y1 6 Y2 ) 2d  A⊆Sd ;|A|>0 `6∈A  X    S \A  P YA1 < YA2 , Y1d  S \A  6 Y2d    +  P(Y1 6 Y2 )  A⊆Sd ;|A|>0  X  2d A⊆S ;|A|>0 d  P YA1 < YA2 +    P(Y1 6 Y2 ) 2d  2d  ,  where the last equality uses the fact that F`−1 (U ) < F`−1 (V ) for any ` ∈ Sd implies that F`−0 1 (U ) 6 F`−0 1 (V ) for all `0 6= ` ∈ Sd . Moreover, P(Y1 6 Y2 ) = P(Y2 > Y1 ) = 1 +  X   (−1)|A| P YA2 < YA1 .  A⊆Sd ;|A|>0  M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  2405  Since P(YA2 < YA1 ) = P(YA1 < YA2 ), 1  T P (Y1 < Y2 ) =  2d 1  =  2d  + +  1  X  1 + (−1)|A| P YA1 < YA2    2d A⊆S ;|A|>0 d 1  X  2d−1  |A|>0 even    P YA1 < YA2 .    As a consequence, 1  Q̃(Md , Md ) = 2T P(Y1 < Y2 ) =  2d−1  +  1  X  2d−2  |A|>0 even  P YA1 < YA2 .    On the other side, P YA1 < YA2 6 min P F`−1 (U ) < F`−1 (V ) = min P (X1` < X2` ) = min D` . `∈A `∈A `∈A      This yields  τ̃d (H ) 6  2  X  min D` . 2d−1 − 1 |A|>0 even `∈A  In order to conclude, it suffices to observe that  X |A|>0 even  d−1 X  min D` = `∈A  D(`) × card {A ⊂ {` + 1, . . . , d}, |A| even}  `=1 d−1 X  =  D(`) ×  `=1 d−1 X  =  1  card {A ⊂ {` + 1, . . . , d}}  2  D(`) × 2d−`−1 ,  `=1  hence the result.    In the identically distributed case, i.e. F1 = · · · = Fd , one has D1 = · · · = Dd and simple computations yield id τ̃max =  d−1  ` X 1  2d D1  2d−1 − 1 `=1  2  =  2d D1   1−  2d−1 − 1  1    2d−1  = 2D1 .  When the variables are continuous, τ̃max = 1 since then, D` = 1/2 for all ` ∈ Sd . 5.2. A bound for Spearman's rho An upper bound for ρ̃d (H ) is established next. Proposition 6. For all H ∈ Γ (F1 , . . . , Fd ), d+1  ρ̃d (H ) 6 ρ̃max =  2d  X  Y n  |A| o1/|A| E ξ` ,  − d − 1 |A|>0 even `∈A  where ξ` = F` (X`− ) + F` (X` ) − 1, ` ∈ Sd . Proof. First observe that  Q̃(Πd , H ) = EH  (   d Y 1 + ξ` `=1  =  =  1  X  2d A⊆S ;|A|>0 d 1 2d−1  2 EH  +  ) d  Y 1 − ξ` 2  `=1  ( Y  ) ξ` +  Y  `∈A  `∈A  ! X |A|>0 even  EH  Y `∈A  (−ξ` ) +  ξ` +  1 2d−1  .  1 2d−1  (13)  2406  M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  By Hölder's inequality,  ! EH  Y  ξ`  n   |A|  6 E ξ`  `∈A  o1/|A|  .  In view of (11), this ends the proof.    √  When d = 2, it is easy to check that ρ̃max = 3{E(ξ12 )}1/2 {E(ξ22 )}1/2 = 3 var(ξ1 )var(ξ2 ), using the fact that E(ξ1 ) = E(ξ2 ) = 0. This is exactly the upper bound derived by Nes̆lehová [9] in the bivariate case. When F1 = · · · = Fd , the random variables ξ1 , . . . , ξd are identically distributed. As a consequence, E(ξ1` ) = · · · = E(ξd` ) for all ` ∈ Sd . Hence,  X  Y n  |A| o1/|A| = E ξ`  |A|>0 even `∈A    X  |A|  E ξ1    |A|>0 even  ! X  =E  | A|  ξ1  |A|>0 even  " =E  X  1 + (−1)|A|  2  A⊆Sd   =E  (1 + ξ1 )d  ξ1 − 1  (1 − ξ1 )d  +  2  # | A|  2   −1 ,  so that  ρ̃  id max  =    d+1  E  2d − d − 1  (1 + ξ1 )d 2  +  (1 − ξ1 )d    −1 .  2  (14)  When the random variables are continuous, ξ1 is stochastically equivalent to 2U − 1, where U is standard uniform. Straightforward computations yield ρ̃max = 1. 5.3. A bound for Spearman's footrule From the proofs of Propositions 5 and 6, one deduces  Q̃(Md , Md ) 6  1 2d−1  +2  d−1  ` X 1  2  `=1  D(`)  and  Q̃(Πd , Md ) =  !  1  X  2d−1 |A|>0 even  EMd  Y  ξ` +  `∈A  1 2d−1  .  A possible bound for Spearman's footrule is then  ϕ̃d (H ) 6 ϕ̃max =    1 2d−1  d+1   (X d−1  d−1  !) 2  d−`  D(`) −  `=1  X  Y  E Md  ξ`  .  (15)  `∈A  |A|>0 even  For identical margins, ξ1 = · · · = ξd under Md , so that  ! X  E Md  Y  ξ`  ! X  =E  `∈A  |A|>0 even  | A|  ξ1   =E  |A|>0 even  (1 + ξ1 )d 2  +  (1 − ξ1 )d 2   −1 .  In that special case, one can then see that  ϕ̃  id max  =  1 2d−1    d+1 d−1    d−1  2  − 1 τ̃   id max   −  2d − d − 1 d+1    ρ̃  id max    .  As is the case for τ̃max and ρ̃max , one can readily check that ϕ̃max = 1 for continuous random variables.  (16)  M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  2407  5.4. Corrected versions Generalizations of Kendall's tau, Spearman's rho and Spearman's footrule are defined next. They are, in fact, corrected versions of τ̃d (H ), ρ̃d (H ) and ϕ̃d (H ) that take into account the bounds in Propositions 5–6 and Eq. (15). As will be seen, they share the enviable property of attaining +1 under a particular kind of perfect association. Definition 1. The normalized versions of Kendall's tau, Spearman's rho and Spearman's footrule are  τ̃d,b (H ) =  τ̃d (H ) , τ̃max  ρ̃d,b (H ) =  ρ̃d (H ) ϕ̃d (H ) and ϕ̃d,b (H ) = . ρ̃max ϕ̃max  A usual condition for a multivariate association measure to be a proper concordance measure is to ensure that its value is +1 under Md . This cannot be fulfilled in the non-continuous case. However, it is possible to weakened this requirement to the case of functionally dependent random variables. This means that X` = G` (X1 ) almost surely, ` ∈ Sd \ {1}, where G2 , . . . , Gd are strictly increasing and continuous functions defined on the range of X1 . Proposition 7. Let HG be the joint distribution of a d-variate vector of functionally dependent random variables. Then  τ̃d,b (HG ) = ρ̃d,b (HG ) = ϕ̃d,b (HG ) = 1. 1 −1 Proof. First note that HG ∈ Γ (F1 , F1 ◦ G− 2 , . . . , F1 ◦ Gd ). In that case, D` = D1 and ξ` = ξ1 for all ` ∈ Sd \ {1}. This situation is then equivalent, from the point of view of the upper bounds, to the identically distributed case. Thus, for this space of id id id distribution functions, τ̃max = τ̃max , ρ̃max = ρ̃max and ϕ̃max = ϕ̃max . It will be shown that τ̃d (HG ), ρ̃d (HG ) and ϕ̃d (HG ) attain these upper bounds. In order to show the result for Kendall's tau, let X1 and X2 be independent and identically distributed random vectors with distribution HG . Hence, for all ` ∈ Sd , X1` = G` (X11 ) and X2` = G` (X21 ) hold almost surely, where G1 (x) = x. One then computes  Q(HG , HG ) = 2P (X1 < X2 ) = 2P {G` (X11 ) < G` (X21 ) , ` ∈ Sd } . By applying the operator T, one obtains  Q̃(HG , HG ) = 2   d  Y T`0 + 1 2  `0 =1  = =  1  X  2d−1 A⊆S d 1  P {G` (X11 ) < G` (X21 ) , ` ∈ Sd }  P {G` (X11 ) 6 G` (X21 ) , ` ∈ A; G` (X11 ) < G` (X21 ) , ` 6∈ A}  2d − 1 P (X11 < X21 ) + P (X11 6 X21 ) .      2d−1  Since P(X11 < X21 ) = D1 and P(X11 6 X21 ) = 1 − P(X21 < X11 ) = 1 − D1 , one finds Q̃(HG , HG ) = {2(2d−1 − 1)D1 + 1}/2d−1 . id Straightforward computations using definition (10) yield τ̃d (HG ) = 2D1 = τ̃max , in view of (13). −1 −1 For Spearman's rho, define Y ∼ Πd ∈ Γ (F1 , F1 ◦ G2 , . . . , F1 ◦ Gd ). One computes  Q(Πd , HG ) = P Y` < F`−1 (U ), ` ∈ Sd + P Y` > F`−1 (U ), ` ∈ Sd    ( = EU  d Y  P Y` <  `=1  F`−1 (U )    +  d Y  P Y` >  `=1  )   F`−1 (U )   .  Hence,  Q̃(Πd , HG ) =  1 2d  " EU  d Y   `=1  P Y` <  F`−1 (U )    +P  Y` 6 F`−1 (U )    +  d Y   P Y` >  `=1  # F`−1 (U )    +P  Y` > F`−1 (U )    .  1 −1 −1 −1 −1 − − Since for all ` ∈ Sd , F` (x) = F1 ◦ G− ` (x) and F` (u) = G` ◦ F1 (u), one finds P{Y` < F` (U )} = F1 ◦ F1 (U ) = F1 (X1 ). By −1 −1 −1 − similar computations, P{Y` 6 F` (U )} = F1 (X1 ), P{Y` > F` (U )} = 1 − F1 (X1 ) and P{Y` > F` (U )} = 1 − F1 (X1 ). Hence,  Q̃(Πd , HG ) =  =  1 2d 1 2d  E  h  F1 (X1− ) + F1 (X1 )  d  i  d + 2 − F1 (X1− ) − F1 (X1 )  E (1 + ξ1 )d + (1 − ξ1 )d .    id In view of (11), ρ̃d (HG ) = ρ̃max . Following the same lines as for Kendall's tau and Spearman's rho, one can show that id ϕ̃d (HG ) = ϕ̃max .   2408  M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  6. Empirical issues 6.1. Sample versions One of the main advantages of using the operator T described in (6) is the possibility of defining easily computable empirical versions for the multivariate measures of concordance encountered so far in the paper. To this end, let Xi = (Xi1 , . . . , Xid ), i ∈ {1, . . . , n}, be a random sample from an arbitrary distribution function H ∈ Γ (F1 , . . . , Fd ). In the continuous case, the sample version of Kendall's tau is based on the estimation of Q(H , H ) provided by the U-statistic  n  −1 X    Qn (H , H ) = 1 Xi < Xj + 1 Xj < Xi , 2 i<j where 1 is the indicator function. Hence, an estimation of Q̃(H , H ) is simply given by Q̃n (H , H ) = T Qn (H , H ), where it is understood that T` 1 (X1 < X2 ) = 1 (X11 < X21 , . . . , X1` 6 X2` , . . . , X1d < X2d ) ,  ` ∈ Sd .  In order to estimate Spearman's rho, first note that the random vector (X11 , X22 , . . . , Xdd ) is distributed as Πd Γ (F1 , . . . , Fd ). Hence, Q(Πd , H ) may be estimated by  Qn (Πd , H ) =     −1  n d+1  X    1  Xi1 ,i1 , . . . , Xid ,id < Xid+1 + 1      Xi1 ,i1 , . . . , Xid ,id > Xid+1      ∈  ,  i1 <···<id+1  which is a U-statistic of degree d + 1. An estimation of Q̃(Πd , H ) is then Q̃n (Πd , H ) = T Qn (Πd , H ). Hence, easily computable empirical versions of τ̃d (H ) and ρ̃d (H ) are 2d−1 Q̃n (H , H ) − 1 2d−1  and ρ̃n,d (H ) =  d+1  2d−1 Q̃n (Πd , H ) − 1 . −1 −d−1 The asymptotic behavior of τ̃n,d (H ) has already been obtained by Quessy [10]. The result is recalled next, together with the asymptotic behavior of ρ̃n,d (H ). Note that the estimation of ϕ̃d is a difficult problem, even in the continuous case. This issue  τ̃n,d (H ) =    2d  should be tackled in a separate investigation. Proposition 8. As n → ∞,  σ̃τ ,d = 2  2  2d    2d−1 − 1  √  n{τ̃n,d (H ) − τ̃d (H )}  N (0, σ̃τ2,d ) and  √  n{ρ̃n,d (H ) − ρ̃d (H )}  2 N (0, σ̃ρ, d ), where  var TH (X) + TH̄ (X)    and  σ̃ρ,d = 2    d+1  2  2d − d − 1  var  (   d Y 1 + ξ` `=1  2  +   d  Y 1 − ξ` `=1  2  ) .  Proof. From classical results in the theory of U-statistics that one can find, for example, in [5], n 1 X n Q̃n (H , H ) − Q̃(H , H ) = √ T Kτ (Xi ) + oP (1) n i=1  √  and  n 1 X n Q̃n (Πd , H ) − Q̃(Πd , H ) = √ T Kρ (Xi ) + oP (1), n i =1  √   where for X ∼ H, Kτ (x) = P(X < x) + P(X > x) and Kρ (x) =  d Y  F` (x` ) +  `=1  d Y  {1 − F` (x` )} .  `=1  The results follows from the central limit theorem and straightforward computations for the asymptotic variances.    In the special case of multivariate independence, one can show that 2 σ̃ρ, d =  1 2d−1    d+1 2d − d − 1  2 ( Y d `=1  1 + σ` +   2  d Y  `=1  ) 1 − σ` − 2 ,   2  where σ`2 = var(ξ` ), ` ∈ Sd . The formula simplifies to 9σ12 σ22 when d = 2. These expressions may be used to define a test of multivariate independence. One only has to estimate σ12 , . . . , σd2 in order to obtain an estimate of σρ2 which is valid under any alternatives. This was the approach taken by Quessy [10] in order to evaluate the small sample and asymptotic power of procedures for multivariate independence.  M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  2409  6.2. Confidence intervals Usually, if one wants to infer on the values of τ̃d and ρ̃d , one needs accurate, nonparametric estimations of σ̃τ ,d and σ̃ρ,d . A solution is to compute jackknife estimates σ̂τ ,d and σ̂ρ,d . Hence, confidence intervals are given by  σ̂τ ,d τ̃d = τ̃n,d ± zα/2 √  and  n  σ̂ρ,d ρ̃d = ρ̃n,d ± zα/2 √ ,  (17)  n  where zα/2 is the (1 − α/2)th percentile of the standard normal distribution. The corrected versions of Kendall's tau and Spearman's rho encountered in Definition 1 can also be estimated from the data. To this end, simply note that D` = |A| E{F` (X`− )} and E(ξ` ) can be estimated, respectively, by  D̂` =  1 2  −  1  n  −1 X 2 2  1 Xi` = Xj`    i<j  and    |A|  Ê ξ`    =  n  1 X |A| Fn` Xi− , ` + Fn` (Xi` ) − 1 n i=1  where Fn1 , . . . , Fnd are the univariate empirical distribution functions. 6.3. Illustration on real data Consider the horseshoe crab data set in Table 3.2 of [1]. The latter consists of n = 173 female crabs, each having a male attached in her nest. The variables that were considered are X1 : color of the female, X2 : spine condition and X3 : number of other males around the female, where X1 ∈ {light med, med, dark med, dark}, X2 ∈ {both good, one good, both bad} and X3 is a count variable. Kendall's tau for these 173 observations is computed as τ̃n,d = .048, while the jackknife estimate of its standard deviation is σ̂τ ,d = .361. In view of (17), a 95% confidence interval is τ̃d = .048 ± .104. In order to estimate the corrected version, note that when d = 3, τ̃max = (4/3)D(1) + (2/3)D(2) . Simple computations yield (D̂1 , D̂2 , D̂3 ) = (.327, .273, .318), so that the upper bound is estimated by τ̂max = (4/3)(.273) + (2/3)(.318) = .576. Hence, τ̃d,b = (.048 ± .104)/.576 = .083 ± .181. For Spearman's rho, one computes ρ̃n,d = .100 and the standard deviation is estimated by σ̂ρ,d = .519. A 95% confidence interval is then ρ̃d = .100 ± .150. When d = 3, ρ̃max will be estimated by ρ̂max =  q  q  Ê(ξ12 )Ê(ξ22 ) +  q  Ê(ξ12 )Ê(ξ32 ) +  Ê(ξ22 )Ê(ξ32 ). Since Ê(ξ12 ) = .285, Ê(ξ22 ) = .246 and Ê(ξ32 ) = .265, one finds ρ̂max = .794. The confidence interval for the  corrected version is then ρ̃d,b = (.100 ± .150)/.794 = .126 ± .189. Both for the trivariate Kendall's tau and Spearman's rho, one sees that 0 is a plausible population value. Hence, a test for the simultaneous association of the three variables would accept the null hypothesis of independence at the 5% level. If, on the contrary, the dependence had been significantly larger than 0, then one could have concluded that the three variables have a tendency to be large (or small) simultaneously, as a consequence of the definition of concordance. For the horseshoe crab data, that would mean that observing a female having a dark color, many males around her and a spine in bad condition has a high probability. The same could be said about females with a light color, few males around and a spine in good condition. 7. Closing remarks The properties of the concordance function Q̃ entails enviable properties for any association measure based on it. For example, one could consider pairwise versions of concordance measures. To this end, let κ̃d (H ) be a given measure of concordance and define its average pairwise version by  κ̃d,pair (H ) =    −1 X d  2   0 κ2 H `` ,  `<`0  ``0  where H is the joint distribution of the pair (X` , X`0 ). For Kendall's tau, Spearman's rho and Spearman's footrule, one can show easily that τ̃d,pair (H ) = 2Q̃pair (H , H ) − 1, ρ̃d,pair (H ) = 6Q̃pair (Π , H ) − 3 and ϕ̃d,pair (H ) = 3Q̃pair (H , M ) − 2, where  Q̃pair (H1 , H2 ) =    −1 X d  2    Q̃ H1`` , H2`` 0  0    .  `<`0  Hence the bounds in Propositions 5–6 and (15) could be used to bound the above pairwise versions. Then, straightforward computations yield  2410  M. Mesfioui, J.-F. Quessy / Journal of Multivariate Analysis 101 (2010) 2398–2410  τ̃d,pair (H ) ≤    −1 X d  2  min {2D` , 2D`0 } =  `<`0    −1 X d−1 d  2  (d − `)2D(`)  `=1  and  ρ̃d,pair (H ) ≤    −1 X p d 2  3 var(ξ` )var(ξ`0 ).  `<`0  The upper bound for ϕ̃d,pair (H ) is more complicated since EM2 (ξ` ξ`0 ) cannot be evaluated explicitly. Acknowledgments Funding in partial support of this work was provided by the Natural Sciences and Engineering Research Council of Canada and by the Fonds Québécois de Recherche sur la Nature et les Technologies. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]  A. Agresti, An Introduction to Categorical Data Analysis, Wiley-Blackwell, 2007. M. Denuit, P. Lambert, Constraints on concordance measures in bivariate discrete data, J. Multivariate Anal. 93 (1) (2005) 40–57. C. Genest, J. Nešlehová, A primer on copulas for count data, Astin Bull. 37 (2) (2007) 475. H. Joe, Multivariate concordance, J. Multivariate Anal. 35 (1) (1990) 12–30. A.J. Lee, U-statistics: Theory and Practice, in: Statistics: Textbooks and Monographs, vol. 110, Marcel Dekker Inc., New York, 1990. M. Mesfioui, A. Tajar, On the properties of some nonparametric concordance measures in the discrete case, J. Nonparametr. Stat. 17 (5) (2005) 541–554. R.B. Nelsen, Concordance and copulas: a survey, in: Distributions with Given Marginals and Statistical Modelling, Kluwer Acad. Publ., Dordrecht, 2002, pp. 169–177. R.B. Nelsen, An introduction to copulas, in: Springer Series in Statistics, second ed., Springer, New York, 2006. J. Nešlehová, On rank correlation measures for non-continuous random variables, J. Multivariate Anal. 98 (3) (2007) 544–567. J.-F. Quessy, Tests of multivariate independence for ordinal data, Comm. Statist. Theory Methods 38 (2009) 3510–3531. C. Spearman, The proof and measurement of association between two things, Am. J. Psychol. 15 (1904) 88. M.D. Taylor, Multivariate measures of concordance, Ann. Inst. Statist. Math. 59 (4) (2007) 789–806. M. Úbeda-Flores, Multivariate versions of Blomqvist's beta and Spearman's footrule, Ann. Inst. Statist. Math. 57 (4) (2005) 781–788. E.F. Wolff, n-dimensional measures of dependence, Stochastica 4 (3) (1980) 175–188.
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Concordance Measures for Multivariate Non continuous Random Vectors
