A survey of copula based measures of association Carlo Sempi 1

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1 Lecture Notes of Seminario Interdisciplinare di Matematica Vol. 1(15), pp A survey of copula based measures of association Carlo Sempi 1 Abstract. We survey the measures of association that are based on bivariate copulas. Almost no proof will be reported, although an exception is made in the case of the Schweizer Wol measure, since the details of the proof are mainly contained in Wol s Ph.D. dissertation, which is not readily available. 1. Introduction Measure of association is a broad term that denotes the class of all the measures that have been constructed with the aim of quantifying specific relationships between two or more random variables. The term may include, for instance, measures that want to capture functional relationship (i.e., linear relationships) among random variables, as well as measures of dependence that aim at quantifying the degree of non independence in a set of variables. In this paper we review those among these measures that may expressed in terms of the copulas of two random variables. In the next section we recall, without any proof, the properties of (bivariate) copulas that will be needed in the sequel. References on copulas are the following books and surveys [9, 14, 4, 5].. Copulæ A (bivariate) copula is a distribution function (=d.f.) on I whose univariate marginals are uniformly distributed on I. Equivalently, a copula C is a function C : I! I such that (C1) C(t, 1) = C(1,t)=t and C(t, ) = C(,t) = for every t I; (C) for all u, u, v and v in I with u apple u and v apple v, C(u,v ) C(u,v) C(u, v )+C(u, v). The set of copulas will be denoted by C. The following three examples of copulas are essential: the comonotonicity copula M (u, v) =min{u, v}, theindependence copula (u, v) =uv and the countermonotonicity copula W (u, v) = max{,u+ v 1}. 1 C. Sempi, Università del Salento, Dipartimento di Matematica e Fisica Ennio De Giorgi, Via per Arnesano, 731 Lecce, Italy; carlo.sempi@unisalento.it Keywords. Measures of association, copulæ, Sperman s rho, Kendall s tau, Schweizer Wol measure. AMS Subject Classification. Primary 6H, Secondary 6E5.

2 Carlo Sempi Other important families of copulas are listed below. Archimedean (.1) C(u, v) =f ( 1) (f(u)+f(v)), where f : I! [, +1] is continuous, convex, strictly decreasing and such that f(1) = and f ( 1) (t) =f 1 (t) ift [,f()], while f ( 1) (t) = for t f(). Eyraud Farlie-Gumbel Morgenstern (briefly EFGM) (.) C EFGM (u, v) =uv (1 + (1 u)(1 v)), [ 1, 1] ; Marshall Olkin for, ], 1[ (.3) C MO, (u, v) :=min u 1 v, uv 1 = ( u 1 v, u v, uv 1, u apple v, Gumbel Hougaard (.4) C GH (u, v) =exp (( ln u) (ln v) ) 1/, 1. For = 1 one obtains the independence copula as a special case, and the limit of C GH for! +1 is the comonotonicity copula M. Bivariate Gaussian (.5) C Ga (u, v) = = 1 1 (u) 1 1 (v) 1 s p st + s 1 (1 ds dt, ) where is in ] 1, 1[, and 1 denotes the inverse of the standard Gaussian distribution N(, 1). For more details, see [1]. Mardia Takahashi Clayton (.6) C MTC (u, v) = ( 1 = max, u + 1 ) 1/ v 1, [ 1+1[ \{}. Frank (.7) C Frank (u, v) = = 1 ln 1+ (e u 1) (e v 1) (e 1), R. The limiting case = corresponds to. Copulas of this type have been introduced by [6] in relation with a problem about associative functions on I. They are absolutely continuous. Extreme value (.8) C A (u, v) = ln v =exp A ln u +lnv ln(uv), (u, v) ], 1[,

3 Copula based measures of association 3 where A : I! [1/, 1] is convex and satisfies the inequality max{1 A(t) apple 1. The importance of copulas stems form Sklar s theorem [3]. t, t} apple Theorem.1 (Sklar s theorem). Let F be a d.f. with univariate margins F 1, F. Let A j denote the range of F j, A j := F j (R) (j =1, ). Then there exists a copula C such that for all (x 1,x ) R, (.9) F (x 1,x )=C (F 1 (x 1 ),F (x )). Such a C is uniquely determined on A 1 A and, hence, it is unique when both F 1 and F are continuous. Conversely, if C is a copula, then (.9) defines a d.f.. If the marginals F 1 and F are continuous the existence of the (unique) copula asserted by Sklar s theorem can be easily proved. Under the assumptions of Theorem.1, if F 1 and F are continuous, then there exists a unique copula C associated with X =(X 1,X ). It is determined, for every (u, v) I d, via the formula (.1) C(u, v) =H F ( 1) 1 (u),f ( 1) (v), where, for j =1,, F ( 1) j (t) =inf{x R : F j (x) t} is the right continuous quasi inverse of F j. Example.1 (Copula of the extreme order statistics). In the probability space (, F, P) letx 1,, X n be indipendent and identically distributed random variables with a common continuous d.f. F and let X (j) be the j th order statistic so that X (1) apple X () apple applex (n). We wish to determine the copula of the vector (X (1),X (n) ). It is known (see, e.g., [3]) that the joint d.f. of the j th and the k th order statistics is H j,k (x, y) = nx hx h=k i=j for x<y, and F (k) (y), for x k th order statistic n! i!(h i)!(n h)! F i (x)[f (y) F (x)] h i [1 F (y)] n h, F (k) (t) = y, wheref (k) (t) :=P(X (k) apple t) is the d.f. of the nx i=k n F i (t) [1 F (t)] n i. i Hence, setting first j = 1 and then k = n, one has, for x and y in R, F (1) (x) =1 [1 F (x)] n, F (n) (y) =F n (y) ; now the joint d.f. of X (1) and X (n) is given by 8 nx >< n F i (x) [F (y) F (x)] n i, x < y, H 1,n (x, y) = i i=1 >: F (n) (y), x y, = ( F n (y) [F (y) F (x)] n, x < y, F n (y), x y.

4 4 Carlo Sempi By recourse to (.1), one can now write the (unique) copula of X (1) and X (n) C 1,n = H 1,n F 1 (1) (u),f 1 (n) (v) = (.11) ( v v 1/n +(1 u) 1/n 1 n, 1 (1 u) 1/n <v 1/n, = v, 1 (1 u) 1/n v 1/n. The copula of eq. (.11) is related to a copula from the Mardia Takahasi Clayton copulas family (.6). In fact, by using the symmetry (u, v) =(1 u, v), one has C 1,n (u, v) = ( v 1/n + u 1/n 1 n, v 1/n + u 1/n 1 >,, elsewhere, which belongs to the Mardia Takahasi Clayton family with = 1/n. One often considers, along with a copula C its survival copula defined by (.1) b C(u, v) :=u + v 1+C(1 u, 1 v). One of the important properties of copulas is contained in the following result. Theorem.. Let X and Y be continuous random variables defined on the probability space (, F, P) and consider the continuous mappings ' : Ran X! R and : Ran Y! R. (a) If both ' and are strictly increasing, then, for every (u, v) I, C '(X), (Y ) (u, v) =C XY (u, v) ; (b) if ' is strictly increasing while is strictly decreasing, then, for every (u, v) I, C '(X), (Y ) (u, v) =C (u, v) :=u C XY (u, 1 v) ; (c) if ' is strictly decreasing while is strictly increasing, then, for every (u, v) I, C '(X), (Y ) (u, v) =C 1 (u, v) :=v C XY (1 u, v) ; (d) if both ' and are strictly decreasing, then, for every (u, v) I, C '(X), (Y ) (u, v) =C 1 (u, v) :=u + v 1+C XY (1 u, 1 v). There is a one to one correspondence between copulas and stochastic measures on (I, B(I )): a measure µ on this measurable space is said to be a stochastic measure if µ(a I) =µ(i A) = (A) for every Borel subsest A of I; here and in the following d denotes the d dimensional Lebesgue measure (d =1, ). Given a copula C for every rectangle R =]a, b] ]c, d] one defines µ C (R) :=C(b, d) C(a, d) C(b, c)+c(a, c) by the usual techniques of measure theory the definition of µ C is then extended to the family B(I ) of Borel subsets of I. Conversely, if a stochastic measure µ is given, a copula C is defined via C(u, v) :=µ([,u] [,v]). Because of this correspondence one may write R dc in order to denote the integral R dµ I I C. The integration by parts formula presented below (see [11]) is needed in the calculation of some of statistical quantities surveyed here.

5 Copula based measures of association 5 Theorem.3. Let A and B be copulas, and let the function ' : I! R be continuously di erentiable, i.e., ' C 1 (I). Then (.13) ' AdB = I 1 = '(t) dt ' (A(u, 1 A(u, B(u, v) du dv = I 1 (.14) = '(t) dt ' (A(u, A(u, 1 B(u, v) du dv. I Now, let P be a property of association that may be assigned to a random vector X =(X 1,X ). A (copula based) measure of association is any functional M P : C! D R that assigns to each vector X of continuous r.v. s with copula C a real number that is interpreted as the P degree of association of X. If no confusion arises, M(X) or, equivalently, M(C) will denote the value of M for a vector X with copula C. Usually, the range D of a measure of association M is either D =[, 1], where M(X) = represents the absence of property P in X or D =[ 1, 1], where M(X) = 1(respectively, M(X) = 1) represents the maximal positive (respectively, negative) presence of property P in X. 3. Concordance Loosely speaking, one may say that two random variables are concordant if they tend to take large values together or to take small values together. Negative concordance of two random variables means that one of them takes large values while the other one takes small values. Let X and Y be two continuous random variables defined on the probability space (, F, P). Given two di erent observations (x j,y j ) and (x k,y k ) of the random vector (X, Y ) these will be said to be concordant, if one has either x j <x k and y j <y k, or, x j >x k and y j >y k, or equivalently, if (x k x j )(y k y j ) > ; discordant, ifx j <x k and y j >y k, or x j >x k and y j <y k, or, equivalently, if (x k x j )(y k y j ) <. Definition 3.1. Given two copulas A and B in C, B will be said to be more concordant than A, and this will be denoted by A B, ifa(u, v) apple B(u, v) for every (u, v) I. In order to investigate concordance between two random pairs, it is expedient, following [1] and [14, Section 5.1.1], to introduce a concordance function Q. Assume that the pair (X 1,Y 1 ) and (X,Y ) of continuous random vectors have (not necessarily equal) joint d.f. s H 1 and H, respectively, but common marginals F and G; thus, both H 1 and H belong to the Fréchet class (F, G), which is the set of two dimensional d.f. s whose marginals are F and G. It will turn out that Q depends only on the copulas C 1 and C of the two vectors. Theorem 3.1. Let X 1, Y 1, X, Y be continuous random variables on the probability space (, F, P), let the random vectors (X 1,Y 1 ) and (X,Y ) be independent and let H 1 and H be their respective joint d.f. s and let the marginals d.f. s satisfy F X1 = F X = F, and F Y1 = F Y = G,

6 6 Carlo Sempi so that H 1 and H both belong to the Fréchet class (F, G), and H 1 (x, y) =C 1 (F (x),g(y)) and H (x, y) =C (F (x),g(y)), where C 1 and C are the (unique) copulas of (X 1,Y 1 ) and (X,Y ), respectively. Define (3.1) Q := P [(X 1 X )(Y 1 Y ) > ] P [(X 1 X )(Y 1 Y ) < ]. Then Q depends only on C 1 and C and is given by (3.) Q(C 1,C )=4 C (u, v) dc 1 (u, v) 1. I Some of the properties of the concordance function Q will be useful in the sequel. They are collected in the following result, whose proof is immediate. Theorem 3.. Let C 1, C and Q have the same meaning as in Theorem 3.1; then: (a) Q is symmetric: Q(C 1,C )=Q(C,C 1 ); (b) Q is increasing in each place with respect to the concordance order : if C 1 C1 and C C, then Q(C 1,C ) apple Q(C1,C ); (c) Q is invariant under the replacement of copulas by their survival copulas: Q(C 1,C )=Q(Ĉ1, Ĉ). Example 3.1. The function Q will be evaluated for all possible pairs of the three fundamental copulas M, and W. As will be seen below (see Section 6), one is at the same time calculating Kendall s for all pair of random variable having M, or or, again, W as their copula. In every case, use will be made of (.13). Since ( ( 1, s < t,, s < 1 M (s, t) = M (s, t) =, s > t, 1, s > t, one has 1 M (s, t) =1 (s,1) (t) =1 (,t) (s), M (s, t) =1 (,s) (t) =1 (t,1) (s). Similarly, so 1 W (s, t) =@ W (s, t) = (, s+ t 1 <, 1, s+ t 1 >, 1 W (s, t) =@ W (s, t) =1 (t 1,1) (s) =1 (1 s,1) (t). As a consequence of Theorem.3 and because of (3.3), one has Q(M,M )=4 M (u, v) dm (u, v) 1= 1=1. I

7 Copula based measures of association 7 Similarly Q(M, ) = Q(,M )=4 Q(M,W )=Q(W,M )=4 Q(W, ) = Q(,W )=4 Q(W,W )=4 Finally Q(, ) = / 1 u du 1= 4 3 (u 1) du 1= u (1 u) du = 1 3, W (u, 1 u) du 1= 1. 1= 1 3, Since Q is the di erence of two probabilities (see eq. (3.1)), one has Q(C, C) [ 1, 1] for every copula C. In view of Theorem 3. (b) and of the previous example, one has also, for every copula C, Q(C, M ) [, 1], Q(C, W ) [ 1, ], Q(C, ) [ 1/3, 1/3]. 4. Spearman s rank correlation Charles Spearman, a psychologist, introduced his rank correlation coe cient, known as Spearman s rho in 194 [4]. In our setting, his coe cient is defined as the normalised concordance between a copula C and the independence copula, so that it measures the discrepancy of concordance property of (X, Y ) with respect to an independent coupling (X,Y) belonging to the same Fréchet class. Definition 4.1. If C is the copula of two continuous random variables X and Y defined on the same probability space (, F, P), then the population version of Spearman s rho of X and Y, which will be denoted indi erently by X,Y or by C or by (C), is given by (4.1) X,Y = (C) =3Q(C, ) = 1 = 1 C(u, v) du dv 3. I I The coe cient 3 that appears in eq. (4.1) represents a normalisation: in fact, thanks to it, C is in [ 1, 1] since Q(C, ) belongs to the interval [ 1/3, 1/3], as was seen in the previous Section. Notice also that (u, v) du dv = uvdudv = 1 I I 4, so that Spearman s rho may be written in the form uv dc(u, v) 1/4 I (4.) (X, Y ) = 1 {C(u, v) uv} du dv = I 1/1 Below the values of Spearman s rho for a few copulas of Section are reported. EFGM copulas (.): (C EFGM )= /3;.

8 8 Carlo Sempi Marshall Olkin copulas (.3): (C, MO )=(3 )( + ) 1 ; Gaussian bivariate copula (.5): (C Ga )= 6 arcsin ; for a proof see [1]; Frank copulas (.7): (C Frank )=1 1 (D 1( ) D ( )), where D n is the Debye function given, for any natural n by (4.3) D n (x) = n x n x Extreme value copulas (.8): t n e t 1 dt ; 1 1 (C A ) = 1 (1 + A(t)) dt 3. Example 4.1. Spearman s rho (X (1),X (n) ) for the copula of the order statistics X (1) and X (n) of the independent and identically distributed random variables X 1,, X n, a copula that was determined in Example.1 is given by (4.4) (X (1),X (n) )=3 1n n n nx k= ( 1) k n + 1 (n!)3 n k n + k (3n)! ( 1)n. It follows from Theorem 4.1 that where n = (X (1),X (n) ) = 1 C n (u, v) du dv 3= I n h i n o = 1 v v 1/n +(1 u) 1/n 1 du dv + 1 vdudv 3= A n B n h n = 1 vdudv 1 v 1/n +(1 u) 1i 1/n du dv 3, A n[b n A n A n := (u, v) I :1 (1 u) 1/n <v 1/n, B n := (u, v) I :1 (1 u) 1/n >v 1/n.

9 Copula based measures of association 9 The substitutions s =(1 u) 1/n and t = v 1/n, and the use of the beta function yield 1 1 (1 v 1/n ) n h n I n = dv v 1/n +(1 u) 1i 1/n du = = n 1 t n 1 1 dt (t + s 1) n s n 1 ds = 1 t = n n X k= k= n 1 1 (t 1) k t n 1 dt s n k+n 1 ds = k 1 t X n n 1 apple = n (t 1) k t n 1 1 k n k 1 (1 t)n k dt = X n n ( 1) = n k apple 1 t n 1 (1 t) k dt k n k k= 1 t n 1 (1 t) n dt = Now = n n X k= B(n, n + 1) n n ( 1) k {B(n, k + 1) B(n, n + 1)}. k n k n X k= n ( 1) k k n k = (n 1)!(n)! (3n)! n n X k= n ( 1) k k n k = Similarly which proves the assertion. B(n, k + 1) = = n(n)!n! (3n)! (n 1)! k! (n + k)! ( 1) n n n n, =( 1) n (n!)3 (3n)!. 5. Gini s cograduation index Spearman s rho of two continuous random variables having C as their copula may be expressed in the form (C) = 1 (u + v 1) (u v) dc(u, v). I In fact, the r.h.s. reduces to 1 uv dc(u, v)+3 6 udc(u, v) 6 vdc(u, v) ; I I I but, since one integrates with respect to the stochastic measure µ C induced by the copula C, 1 6 udc(u, v) =6 udu=3, I

10 3 Carlo Sempi so that the claimed equality holds. Gini s index replaces the square by the absolute value, and, of course, adopts a di erent normalizing constant (5.1) (C) := { u + v 1 u v } dc(u, v). I Gini s index may also be expressed in terms of the concordance function. Theorem 5.1. Gini s cograduation index for a copula C C may be expressed in either of the following forms 1 =4 C(u, 1 (C) =Q(C, M )+Q(C, W )= u) du 1 (u C(u, u)) du. 6. Kendall s rank correlation Kendall s tau is defined as the concordance between a random pair (X, Y ) and an independent copy of it. Formally, it is defined as follows. Definition 6.1. The Kendall s tau of two continuous random variables X and Y on the probability space (, F, P) isdefinedby (6.1) X,Y = Q(C, C) =4 C(u, v) dc(u, v) 1. I In view of (.13), the expression for may be written as (6.) X,Y =1 1 C(u, C(u, v) du dv. I Since Kendall s tau for a pair of continuous random variables X and Y having copula C, depends only on the copula C, so that, in order to stress this dependence, we shall write X,Y = C = (C). The result of Example 3.1 yield (M )=Q(M,M )=1, ( )=, (W )= 1. The calculation of Kendall s tau is easier for Archimedean copulas since in this case one has to evaluate the integral of a function of a single variable, the generator of the copula, rather than the integral of functions of two variables. Theorem 6.1. Kendall s tau (C f ) for an Archimedean copula C ' with additive generator f is given by (6.3) (C f )=1+4 1 f(t) f (t) dt. Below, the values of Kendall s tau for a few copulas are reported. EFGM copulas (.): (C EFGM )= /9; Marshall Olkin copulas (.3): (C MO, )= + ; Gumbel Hougaard copulas (.4): (C GH )=( 1)/ ; Mardia Takahasi Clayton copulas (.6): (C MTC )= /( + );

11 Frank copulas (.7): (C Frank )=1 function of eq. (4.3); Extreme value copulas (.8): Copula based measures of association 31 (C A )= 1 4 (1 t(1 t) A(t) da (t). D 1( )), where D 1 is the Debye The EFGM copulas do not allow the modelling of a large spectrum of dependence among the random variables involved, since (C EFGM ) [ /9, /9], while (C EFGM ) [ 1/3, 1/3]. Moreover, all the tail dependence coe cients, to be met later, see Section 9, associated with it are equal to. Example 6.1. As was done in Example 4.1 for Spearman s rho, one may wish to calculate Kendall s tau (X (1),X (n) ) for the copula of the order statistics X (1) and X (n) of the independent and identically distributed random variables X 1,, X n, whose copula was determined in Example.1. However, the calculation is long and tedious; therefore we shall limit ourselves to quoting the results from the literature ([1]). Kendall s tau for the order statistics X (j) and X (k) (j, k =1,,n; j<k)is given by (X (j),x (k) )=1 nx k Xj 1 (6.4) h= s= n n s (n 1) n 1 s h n j 1 n j 1 k j 1 n n k + h, s + j 1 Setting j = 1 and k = n one has 1 (6.5) (X (1),X (n) )= n 1, a result originally obtained by Schmitz [18, 19]. 7. Blomqvist s rank correlation 1. If in the expression (3.1) for the concordance function one takes X = x and Y = y, namely constant random variables, one obtains Q = P [(X 1 x )(Y 1 y ) > ] P [(X 1 x )(Y 1 y ) < ]. Blomqvist [] chose x = ex and y = ey, whereex and ey denote the medians of X and Y, respectively. The corresponding measure of concordance, which is usually called Blomqvist s beta or medial correlation coe cient is then (7.1) = P [(X ex)(y ey) > ] P [(X ex)(y ey) < ] = =P [(X ex)(y ey) > ] 1=4C(1/, 1/) On the definition of measure of concordance In two papers [16, 17] Scarsini introduced axioms for a measure of concordance. These are collected in the following Definition 8.1. Consider the family c of continuous dimensional distribution functions and let (X, Y ) be a random pair having joint distribution function H c. Then a mapping apple : c! [, 1], denoted by apple(x, Y ) or by apple(h) is said to be a measure of concordance if

12 3 Carlo Sempi (apple1) apple is defined for every pair of continuous random variables; (apple) apple is symmetric: apple(x, Y )=apple(y,x); (apple3) apple is increasing in the sense that if C X,Y C W,,whereC X,Y and C W, are the copulas of the pairs (X, Y ) and (W, ), respectively, then apple(x, Y ) apple(w, ); (apple4) apple(x, Y ) [ 1, 1]; (apple5) apple(x, Y )= ifx and Y are independent; (apple6) apple( X, Y )=apple(x, Y )= apple(x, Y ); (apple7) weak continuity: if {(X n,y n )} nn is a sequence of continuous random variables and if the corresponding sequence of copulas converges pointiwise to the copula C of the pair (X, Y ), then lim apple(x n,y n )=apple(x, Y ). n!+1 If the inequalities in (apple3) are replaced by C X,Y >C W, and apple(x, Y ) >apple(w, ), respectively, then apple is said to be a strong measure of concordance. In terms of copulas it is possible to reformulate Scarsini s axioms in the following way. Definition 8.. A measure of concordance is a mapping apple : C! R such that (apple1) apple is defined for every copula C C ; (apple) for every C C, apple(c) =apple(c T ); (apple3) apple(c) apple apple(c )wheneverc apple C ; (apple4) apple(c) [ 1, 1]; (apple5) apple( )=; (apple6) apple(c 1 )=apple(c )= apple(c), where C 1 and C are defined in Theorem.; (apple7) weak continuity: if C n! n!+1 apple(c n )=apple(c). n!+1 A measure of concordance is invariant under strictly increasing transformations. Theorem 8.1. The following statements hold for a measure of concordance apple: (a) If the continuous functions f,g : R! R are simultaneously strictly increasing or strictly decreasing then apple(f X, g Y )=apple(x, Y ); (b) if f : R! R is continuous and strictly increasing (respectively, decreasing), then apple(x, f X) = 1, respectively apple(x, f X) = 1. Proof. (a) Set W := f X and := g Y. If f and g are strictly increasing, then, by Theorem., one has C X,Y = C W, so that the assertion follows from (apple3). If, on the other hand, f and g are both strictly decreasing, apply what has just been proved to the strictly increasing functions f and g. Then, it follows from (apple6) that apple(x, Y )=apple(( f) X, ( g) Y )= =( 1) apple(f X, g Y )=apple(f X, g Y ). (b) If f is strictly increasing, then C X,f X = M ;then(apple4) yields apple(x, f X) = 1, while if f is strictly decreasing, then C X,f X = W so that apple(x, f X) = 1. The quantities introduced in the previous sections are measures of concordance according to Scarsini s definition.

13 Copula based measures of association 33 Theorem 8.. Spearman s rho, Gini s index, Kendall s tau and Blomqvist s beta are all measures of concordance. Proof. The proof of this result presents no di culty with one exception: in proving that Kendall s tau satisfies the convergence property (apple7) a special argument is needed. Recall that a sequence of finite measures (µ n ) on (R d, B(R d )) is said to converge vaguely to µ if lim fdµ n = fdµ, n!+1 for every continuous function f with compact support. Then the following statements are equivalent (see [1]): (a) lim n!+1 R f n dµ n = R fdµ<+1; (b) lim a!+1 sup nn R{f n>a} f n dµ n =. It su ces now to choose = I, f n = C n and µ n = µ Cn. 9. Tail dependence For bivariate probability distributions it is possible to introduce the notion of tail dependence (see [8]); this is related just to the amount of dependence in the upper right quadrant tail or in the lower left quadrant tail. Their measure is via the tail dependence coe cients that were introduced by Sibuya [] and which are defined below. Definition 9.1. Let X =(X, Y ) be a vector with continuous components and let F and G be the d.f. s of X and Y,respectively.Theupper tail dependence coe cient U of X is defined by (9.1) U := lim P X>F ( 1) (t) Y > G ( 1) (t), t!1 t<1 if this limit exists. X is said to be upper tail dependent when tail independent when U =. The lower tail dependence coe cient L of X is defined by (9.) L := lim P X apple F ( 1) (t) Y apple G ( 1) (t), t! t> if this limit exists. X is said to be lower tail dependent when independent when L =. U > and upper U > and lower tail These coe cients, and, hence, tail dependence, depend only on the copula of X. Theorem 9.1. Let X have copula C C ; if the limits of Definition 9.1 exist, and if C(t) :=C(t, t) denotes the diagonal of C; then (9.3) U =lim t! t> and 1 t + C(t, t) 1 t (9.4) L := lim t! t> C(t, t) t =lim t! t> =lim t! t> 1 t + C (t) 1 t C(t) t.,

14 34 Carlo Sempi Proof. By recourse to the survival copula of (.1) one has, since F (X) and G(Y ) are uniformly distributed on I, P X>F ( 1) (t) Y > G ( 1) (t) = P (F (X) >t G(Y ) >t)= C(t, = b t) 1 t = 1 t + C(t), 1 t which proves (9.3) if the limit exists. In a similar manner one proves (9.4). We report the tail dependence coe Marshall Olkin (.3): cients of a few bivariate copulas L(C MO, ) =, U (C MO, )=min{, }; bivariate Gaussian (.5): L(C Ga )= U (C Ga ) = ; Gumbel Hougaard copulas (.4)): L(C GH ) =, U (C GH )= 1/ ; Mardia Takahasi Clayton copulas (.6): ( 1, >, L(C MTC )=, [ 1, ], U (Cl) =; Frank copulas (.7)): L(C Frank )= U (C Frank ) = (for more details, see [13, 7]); Extreme value copulas (.8): L(C A ) =, U (C A )=(1 A(1/)). 1. The Schweizer Wolff measure of dependence Let X and Y be continuous random variables and let F and G be their d.f. s, H their joint d.f., and C their (unique) connecting copula. The graph of C is a surface over the unit square, which is bounded above by the surface z = M (u, v), and is bounded below by the surface z = W (u, v). If X and Y happen to be independent, then the surface z = C(u, v) is the hyperbolic paraboloid z = uv. The volume between the surfaces z = C(u, v) and z = uv can be used as a measure of dependence. Notice that {M (u, v) uv} d = 1 and {uv W (u, v)} d = 1 I 1 I 1, which, by normalizing, leads to the quantity (1.1) (X, Y ) := 1 C(u, v) uv d = 1 C d. I I This quantity will be called the Schweizer Wol measure of dependence [, 6]: it depends only on the copula C of the continuous random variables X and Y and represents the L 1 distance between the surfaces z = (u, v) and z = C(u, v). We proceed to establish its properties. By a standard change of variables this measure of dependence may be expressed in the form (1.) (X, Y ) = 1 H(u, v) F (u) G(v) df (u) dg(v). I

15 Copula based measures of association 35 Theorem 1.1. For every copula C, (X, Y ) takes values in [, 1]; moreover, (X, Y )=1, if, and only if, either C = M or C = W. Proof. Since, by definition it is obvious that (X, Y ) takes values in [, 1], our attention will be devoted to proving the remaining assertion. Fix v in I and introduce the vertical sections from I into [,v ]definedby Put C (u) :=C(u, v ), W (u) :=W (u, v ), P (u) :=uv m, M (u) :=M (u, v ). A(C ):= 1 C (u) uv du. It will be proved that A(C ) attains its maximum when either C = M or C = W. Since the graphs of W and M form a parallelogram contained in I, of which the graph of P is the diagonal, the equality A(W )=A(M ), is proved. Now we shall establish the inequality (1.3) A(C ) apple A(W )=A(M ). Case 1: C P. Since C apple M and the three functions C, P and M are all continuous, inequality (1.3) follows at once. A similar argument holds when C apple P. Case : C is neither everywhere greater nor everywhere smaller than P. Because of the continuity of both C and P,theset{uI: C (u) <P (u)} is open, and is, therefore the union of at most countably many disjoint open intervals, say [ ii ]r i,s i [. For every i I, letp i denote the point (r i,c (r i )) and q i the point (s i,c (s i )); notice that both p i and q i lie on the diagonal of the parallelogram described above. Through p i and q i draw parallels to the side of the parallelogram and denote by m i and n i the points of intersection. Next, let f i and g i the piecewise linear functions defined on [r i,s i ] and determined by p i, m i, q i and p i, n i, q i,respectively. SinceC is increasing and satisfies the Lipschitz condition, the graph of f i bounds below the portion of the graph of C that lies between r i and s i. Therefore, since C <P on ]r i,s i [, one has (1.4) si r i C (u) uv du apple si si r i f i (u) uv du = = g i (u) uv du. r i Define a on I via ( C (u), if C (u) P (u) := g i (u), if C (u) apple P (u) and r i <u<s i. Then A(C )= X ii si r i C (u) uv du + X ii apple A(@ ) apple A(M ), ri+1 s i C (u) uv du apple

16 36 Carlo Sempi where the last inequality follows from Case 1. Finally, since every copula is continuous, the function v 7! A(C v ) is also continuous on I. As a consequence, since 1 C d = A(C v ) dv, I inequality (1.3) implies that R C I d attains its maximum if, and only if, either C = M or C = W. Theorem 1.. For every pair of continuous random variables X and Y on the probability space (, F, P), (X, Y )= (Y,X). Proof. For every point (u, v) ini, one has C X,Y (u, v) =C Y,X (v, u). Theorem 1.3. For a pair of continuous random variables X and Y on the probability space (, F, P), the following statements are equivalent: (a) (X, Y )=; (b) X and Y are independent. Proof. Since C and are continuous, R C I d = if, and only if, C =. Theorem 1.4. For a pair of continuous random variables X and Y on the probability space (, F, P), the following statements are equivalent: (a) (X, Y )=1; (b) there exist two strictly monotonic functions ', : R! R, such that X = ' Y a.e., and Y = X a.e.. Proof. By Theorem 1.1, (X, Y ) = 1, if, and only if, either C X,Y = M or C X,Y = W. But C X,Y = M, if, and only if, X and Y are a strictly increasing function of each other, while C X,Y = W (see, e.g., [5]), if, and only if, X and Y are a strictly decreasing function of each other. Theorem 1.5. Let X and Y be random variables on the probability space (, F, P) and let ' : Ran X! R and : Ran Y! R be strictly monotonic and such that ' X and Y have continuous d.f. s. Then (1.5) (' X, Y )= (X, Y ). Proof. If both ' and are strictly increasing, then the assertion is an immediate consequence of Theorem. (a). If ' is strictly increasing while is strictly decreasing, Theorem. (b) yields, for every (u, v) I, C ' X, Y (u, v) uv = u C X,Y (u, 1 v) uv = u (1 v) C X,Y (u, 1 v), so that by the change of variables s = u, t =1 v (' X, Y ) = 1 C ' X, Y (u, v) uv du dv = I = 1 st C X,Y (s, t) ds dt = (X, Y ). I

17 Copula based measures of association 37 If ' is strictly decreasing, while is strictly increasing, one has, because of Theorem 1. and of what has just been proved (' X, Y )= ( Y,' X) = (Y,X)= (X, Y ). Finally if both ' and are strictly decreasing, then Theorem. (d) yields, for every (u, v) I, C ' X, Y (u, v) uv = u + v 1+C X,Y (1 u, 1 v) uv = = C X,Y (1 u, 1 v) (1 u)(1 v), so that by the change of variables s =1 u, t =1 v, (' X, Y ) = 1 C ' X, Y (u, v) uv du dv = I = 1 C X,Y (s, t) st ds dt = (X, Y ), I which concludes the proof. In order to evaluate (X, Y ) when the joint d.f. of X and Y is bivariate normal, one needs an auxiliary result, a di erential relation for the bivariate normal density. Lemma 1.1. Let ' be the standard bivariate normal density, namely 1 x ' (x, y) = p 1 exp xy+ y (1. ) Proof. The characteristic function f of ' is given by t f(t 1,t )=exp 1 t 1 t + t. The multivariate inversion formula, e.g., [5, p. 1] yields ' (x, y) = 1 ( ) exp( i (xt 1 + yt )) f(t 1,t ) dt 1 dt = R = 1 t ( ) exp( i (xt 1 + yt )) exp 1 t 1 t + t dt 1 dt, R (x, = 1 ( ) ' (x, y) t ( t 1 t )exp( i (xt 1 + yt )) exp 1 t 1 t + t dt 1 dt, R which proves the assertion. From the definition of d.f. one immediately has

18 38 Carlo Sempi Theorem 1.6. The d.f. (x, of the standard bivariate normal satisfies the relation (x, = ' (x, y), (x, y) R. Corollary 1.1. The d.f. of the standard bivariate normal satisfies for all (x, y) R, the inequalities: (x, y) > (x, y) =F X (x) F Y (y) if >, (x, y) < (x, y) =F X (x) F Y (y) if <. Theorem 1.7. Let X and Y be random variables whose joint d.f. is standard bivariate normal, with correlation coe cient. Then (1.8) (X, Y )= 6 arcsin. Proof. Consider first the case in which >, and X and Y have zero mean and standard deviation equal to 1. By (1.) and Corollary 1.1, one has (X, Y )= 1 x + y {F X,Y (x, y) F X (x) F Y (y)} exp dx dy, R and, in view of Theorem (X, Y ) = X,Y (x, y) x + y exp dx dy 3 x = p xy+ y x + y exp 1 R (1 exp dx dy = ) 3 ( = p ) x + xy+( ) y exp 1 R (1 dx dy. ) Consider the change of variables 4 1/ ( ) x + y s = ( y, t = ) ( (1 )( )), 1/ so that s + t = ( ) x + xy+( ) y (1. ) The Jacobian J of this transformation is s 1 J = 4 so (X, Y Integrating yields 6 = p exp (s + t ) ds dt = 4 R (X, Y )= 6 arcsin + k. 6 p 4.

19 Copula based measures of association 39 The constant k is zero, since for the correlation coe cient of a pair of random variables X and Y with bivariate normal distribution is zero if, and only if, they are independent, namely, if, and only if, (X, Y ) =. Thus, for, (X, Y )= 6 arcsin. Assume now <. Then the joint d.f. of X and Y is bivariate normal with correlation coe cient >; thus, from the last expression and from Theorem 1.5 one has (X, Y )= (X, Y )= 6 arcsin = 6 arcsin. In the general case, let X and Y have means m 1 and m and standard deviations 1 and ; by Theorem 1.5 with '(t) = t m 1 1 and (t) = t m, one has (X, Y )= (' X, Y )= 6 arcsin, which completes the proof. Finally we study the behaviour of the Schweizer Wol measure of dependence with respect to weak convergence. Theorem 1.8. Let {(X n,y n )} nn be a sequence of bivariate random vectors in the probability space (, F, P) and, for every n N, let the joint d.f. H n of (X n,y n ) be continuous. If the sequence {(X n,y n )} n converges in law to the random vector (X, Y ) with continuous joint d.f. H, then lim (X n,y n )= (X, Y ). n!+1 Proof. Let C n be the unique copula of H n, and C the unique copula of H. Then, by dominated convergence, lim (X n,y n )= lim 1 C n d = 1 C d = (X, Y ), n!+1 n!+1 I I whence the assertion. It is now possible to list the properties of the Schweizer Wol measure of dependence : (SW1) is defined for every pair of continuous random variables X and Y defined on the same probability space (, F, P); (SW) is symmetric, (X, Y )= (Y,X); (SW3) for every pair of random variables X and Y defined on the same probability space (, F, P), (X, Y ) belongs to [, 1]; (SW4) (X, Y ) = if, and only if, X and Y are independent; (SW5) (X, Y )= 1ifeitherX = ' Y or Y = X for some strictly monotonic functions ', : R! R; (SW6) if ', : R! R are strictly monotonic and such that ' X and Y have continuous d.f. s. Then (' X, Y )= (X, Y );

20 4 Carlo Sempi (SW7) if the joint distribution of X and Y is a bivariate normal distribution with correlation coe cient, then (X, Y )= 6/ arcsin( /); (SW8) if (X n,y n ) has joint continuous d.f. H n and converges in law to the random vector (X, Y ) with continuous joint d.f. H,then (X n,y n )! (X, Y ). The Schweizer Wol measure of dependence is the normalized L 1 distance on the standard probability space (I, B(I ), ) between the copulas C and. Of course, other norms can be taken into consideration for a pair of continuous random variables X and Y on the same probability space (, F, P): the L 1 norm: (1.9) 1(X, Y ):=k 1 k C k 1 = k 1 sup (u,v)i C(u, v) (u, v) ; the L p norm: p(x, Y ):=k p I C(u, v) (u, v) p d 1/p ; here k 1 and k p are normalizing constants. Next 1 will be briefly considered. First, we determine the constant k 1 of (1.9). Lemma 1.. One has and sup M (u, v) uv =1/4, (u,v)i sup W (u, v) uv =1/4. (u,v)i Proof. SinceM,itsu cestostudythedi erencem.ifu apple v, then M (u, v) (u, v) =u uv apple u u = M (u, u) (u, u), and this quantity assumes its maximum 1/4 at u =1/. Similarly if u apple v, then apple (u, v) W (u, v) apple uv u v +1=(1 u)(1 v) apple (1 u) that takes its maximum 1/4 at u =1/. Thus 1(X, Y ):=4k C k 1 =4 sup (u,v)i C(u, v) (u, v). Since all copulas are continuous on I, the supremum is actually a maximum, so that a point (u,v )existsini such that 1(X, Y )=4 C X,Y (u,v ) (u,v ). Theorem 1.9. For all random variables X and Y defined on the same probability space (, F, P), 1(X, Y ) satisfies properties (SW1) (SW6) and (SW8). Now let X and Y have bivariate normal joint d.f.. We shall need the following lemmata.

21 Lemma 1.3. For the the standard normal d.f. Copula based measures of association 41 one has, sup (x,y)r (x, y) (x) (y) =sup (x,y)r (x, y) (x, y) = = (, ) (, ). Proof. The marginals of are both standard normal d.f. s, so that (, ) = ( ()). From Corollary 1.1, one has (x, y) > (x, y) for >, and (x, y) < (x, y) for < and from Theorem 1.6 (1.1) (x, 1 1 p 1 exp whose maximum is attained at (, ). Lemma 1.4. The d.f. Proof. One has from (1.1) whence, integrating satisfies (x, y) x xy + y (1 ) (, ) = arcsin( (, = 1 p 1, (, ) = 1 arcsin( )+k. The constant k is determined by considering that (, ) = ( ()) =1/4, so that k = 1/4, which establishes the assertion. It is now possible to state, Theorem 1.1. Let X and Y be random variables with standard bivariate normal joint d.f. and with correlation coe cient. Then (1.11) 1(X, Y )= arcsin( ). In 1959, Alfred Rényi [15] proposed a list of axioms for a measure of dependence R of two random variables X and Y defined on the same probability space (, F, P). The properties of the Schweizer Wol measure of dependence are fairly close to Rényi s axioms. The main di erences are that Rényi defined R for any pair of random variables X and Y that are not a.e. constant, that in (SW5) the function ' and were assumed to be Borel measurable rather than strictly monotonic, that in (SW6) ' and were assumed to be one to one and Borel measurable; finally, Rényi required R(X, Y )=, while in (SW7) (X, Y ) is a function of. The relationship between Spearman s and the Schweizer Wol measure of dependence follows from the definition of this latter measure and the expression (4.):

22 4 Carlo Sempi for a pair of continuous random variables X and Y defined on the same probability space (, F, P), (X, Y ) apple (X, Y ). References [1] J. Avérous, C. Genest & S.C. Kochar, On the dependence structure of order statistics, J. Multivariate Anal., 94(5), [] N. Blomqvist, On a measure of dependence between two random variables, Ann.Math. Statist., 1(195), [3] H.A. David, Order statistics, Wiley, New York, [4] F. Durante & C. Sempi, Copula theory: an introduction, In P. Jaworski, F. Durante, W. Härdle and T. Rychlik, eds., Workshop on Copula Theory and its Applications, Lecture Notes in Statistics Proceedings, Springer, Berlin Heidelberg, 198(1), 3 31 (1). [5] F. Durante & C. Sempi, Principles of copula theory, Chapman&Hall/CRC,BocaRaton FL, 15. [6] M.J. Frank, On the simultaneous associativity of F (x, y) and x+y F (x, y), Aequationes Math., 19(1979), [7] C. Genest, Frank s family of bivariate distributions, Biometrika,74(1987), [8] H. Joe, Parametric families of multivariate distributions with given marginals, J.Multivariate Anal., 46(1993), 6 8. [9] H. Joe, Multivariate models and dependence concepts, Chapman&Hall,London,1997. [1] W.H. Kruskal, Ordinal measures of association, J. Amer. Statist. Soc., 53(1958), [11] X. Li, P. Mikusiński & M.D. Taylor, Some integration by parts formulas involving copulas, inc.m. Cuadras, J.Fortiana,J.A.RodríguezLallena,eds.,Distributions with given marginals and statistical modelling, Kluwer,Dordrecht, [1] C. Meyer, The bivariate normal copula, Comm.Statist.TheoryMethods,4(13),4 4. [13] R.B. Nelsen, Properties of a one-parameter family of bivariate distributions with specified marginals, Comm.Statist.A TheoryMethods,15(1986), [14] R.B. Nelsen, An introduction to copulas, ndedition,springer,newyork,6. [15] A. Rényi, On measures of dependence, ActaMath.Acad.Sci.Hung.,1(1959), ; see also P. Turán, ed., Selected papers of Alfréd Rényi, Akadémiai Kiadó, Budapest, 1(1976), [16] M. Scarsini, On measures of concordance, Stochastica,8(1984),1 18. [17] M. Scarsini, Strong measures of concordance and convergence in probability, Riv.Mat. Sci. Econom. Social., 7(1984), [18] V. Schmitz, Copulas and stochastic processes, Ph.D. Dissertation, Rheinische Westfälische Technische Hochschule, Aachen, Germany, 3. [19] V. Schmitz, Revealing the dependence structure between X (1) and X (n),j.statist.plann. Inference, 13(4), [] B. Schweizer & E.F. Wol, On nonparametric measures of dependence for random variables, Ann.Statist.,9(1981), [1] R. Serfozo, Convergence of Lebesgue integrals with varying measures, Sankhyā,44(198), [] M. Sibuya, Bivariate extreme statistics. I, Ann.Inst.Statist.Math.Tokyo,11(196), [3] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publ.Inst.Statist. Univ. Paris, 8(1959), [4] C. Spearman, The proof and measurement of association between two things, Amer.J. Psycol., 15(194), [5] S.S. Wilks, Mathematical statistics, Wiley, New York, 196. [6] E.F. Wol, Measures of dependence derived from copulas, Ph.D.Dissertation,University of Massachusetts, Amherst MA, 1977.