Statistics & Probability Letters 54 (2001) 277–282

Distribution functions of copulas: a class of bivariateprobability integral transforms�

Roger B. Nelsena ;∗, Jos)e Juan Quesada-Molinab, Jos)e Antonio Rodr)/guez-Lallenac,Manuel )Ubeda-Floresc

aDepartment of Mathematical Sciences, Lewis & Clark College, 0615 SW Palatine Hill Road, Portland, OR 97219, USAbDepartamento de Mat(ematica Aplicada, Universidad de Granada, Granada, Spain

cDepartamento de Estad(,stica y Mat(ematica Aplicada, Universidad de Almer(,a, Almer(,a, Spain

Received March 2000; received in revised form September 2000

Abstract

We discuss a two-dimensional analog of the probability integral transform for bivariate distribution functions H1 andH2, i.e., the distribution function of the random variable H1(X; Y ) given that the joint distribution function of the randomvariables X and Y is H2. We study the case when H1 and H2 have the same continuous marginal distributions, showingthat the distribution function of H1(X; Y ) depends only on the copulas C1 and C2 associated with H1 and H2. We examinevarious properties of these “distribution functions of copulas”, and illustrate applications including dependence orderingsand measures of association. c© 2001 Elsevier Science B.V. All rights reserved

MSC: primary 60E05; secondary 62H05; 62E10

Keywords: Copulas; Dependence orderings; Distribution functions; Measures of association; Probability integral transform

1. Introduction

If X is a continuous random variable with distribution function F , then it is an elementary exercise to showthat the random variable U =F(X ) (the probability integral transform of X ) is uniformly distributed on theunit interval I= [0; 1]; or equivalently, P[F(X )6 t] = t for all t in I. In this paper we examine an analogoussituation in two dimensions. Let X and Y be continuous random variables with distribution functions F and G,respectively, and let H1 and H2 be bivariate distribution functions whose univariate margins are F and G (therestriction to common margins for H1 and H2 insures that X and Y will be subject to univariate probabilityintegral transforms). Then H1(X; Y ) is a one-dimensional random variable. If the joint distribution function ofX and Y is H2, what can be said about the distribution function of H1(X; Y )? The answer to this question

� Research supported by the Spanish C. I. C. Y. T. grant (PB98-1010), Lewis & Clark College, and the Junta de Andaluc)/a.∗ Corresponding author.E-mail address: nelsen@lclark.edu (R.B. Nelsen).

0167-7152/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reservedPII: S0167 -7152(01)00060 -8

278 R.B. Nelsen et al. / Statistics & Probability Letters 54 (2001) 277–282

is of interest for at least two reasons. First of all, it leads to orderings on the set of continuous bivariatedistribution functions, and secondly, population versions of rank correlation coeMcients (such as Spearman’srho, Kendall’s tau, the Gini coeMcient, and Spearman’s footrule) can be expressed succinctly in terms ofdistribution functions of this form.Before proceeding, we review some elementary properties of copulas. A (bivariate) copula is a func-

tion C : I2 → I which satisQes (a) C(t; 0)=C(0; t)= 0 and C(t; 1)=C(1; t)= t for every t in I, and (b)C(u2; v2)−C(u2; v1)−C(u1; v2)+C(u1; v1)¿ 0 for all u1; u2; v1; v2 in I such that u16 u2 and v16 v2. Equiv-alently, a copula is the restriction to I2 of a bivariate distribution function whose margins are uniform onI. The importance of copulas in statistics is described in the following theorem (Sklar, 1959). Let X andY be random variables with joint distribution function H and marginal distribution functions F and G, re-spectively. Then there exists a copula C (which is uniquely determined on Range F × RangeG) such thatH (x; y)=C(F(x); G(y)) for all x; y. Thus copulas link joint distribution functions to their one-dimensionalmargins. For further details, see Nelsen (1999).In the next section we show that the distribution function of the random variable H1(X; Y ), given that the

joint distribution function of X and Y is H2, depends only on the copulas C1 and C2 associated with H1 andH2. In particular, like the (one-dimensional) probability integral transform, it is independent of the marginaldistribution functions F and G. However, unlike the probability integral transform, it is rarely uniformlydistributed on I.

2. Distribution functions of copulas

We begin with some notation. As is customary, we let M and W denote the Fr)echet–HoeSding upperand lower bound copulas, respectively, which for any copula C satisfy W (u; v)=max(u+v−1; 0)6C(u; v)6min(u; v)=M (u; v). For continuous random variables X and Y , each of X and Y is almost surely an increasing(decreasing) function of the other if and only if their copula is M (W ). The copula of independent continuousrandom variables is �(u; v)= uv. We denote the distribution function of a random variable X either bydf (X ) or a letter such as F ; and we use “6st ” to denote stochastic inequality, i.e., X 6st Y if and only ifdf (X )¿ df (Y ). We let �H denote the measure on R2 induced by the bivariate distribution function H (andsimilarly �C denotes the measure on I2 induced by the copula C). Finally, �C is the diagonal section of C,given by �C(t)=C(t; t).

De�nition 2.1. Let H1 and H2 be bivariate distribution functions with common continuous marginal distribu-tion functions F and G. Let X and Y be random variables whose joint distribution function is H2, and let〈H1|H2〉(X; Y ) denote the random variable H1(X; Y ). The H2 distribution function of H1, i.e., df (〈H1|H2〉(X; Y )),which we denote (H1|H2), is given by

(H1|H2)(t)=Pr[〈H1|H2〉(X; Y )6 t] = �H2 ({(x; y)∈R2|H1(x; y)6 t}); t in I:

Since copulas are bivariate distribution functions with uniform [0; 1] margins, we have an analogous def-inition for copulas: If C1 and C2 are any two copulas, and if U and V are uniform [0; 1] random variableswhose joint distribution function is C2, then 〈C1|C2〉(U; V ) denotes the random variable C1(U; V ), and the C2

distribution function of C1 is given by

(C1|C2)(t)=Pr[〈C1|C2〉(U; V )6 t] = �C2 ({(u; v)∈ I2 |C1(u; v)6 t}); t in I: (2.1)

The following theorem is a bivariate analog of the probability integral transform.

Theorem 2.1. Let H1; H2; F; G; X; and Y be as in De=nition 2:1; and let C1 and C2 be the copulas asso-ciated with H1 and H2. Then (H1|H2)= (C1|C2).

R.B. Nelsen et al. / Statistics & Probability Letters 54 (2001) 277–282 279

Fig. 1. The distribution functions of 〈C1|C2〉 for C1; C2 ∈{M;�;W}.

Proof. For t in I, we have

(H1|H2)(t) = �H2 ({(x; y)∈R2 |H1(x; y)6 t})= �H2 ({(x; y)∈R2 |C1(F(x); G(y))6 t});= �C2 ({(u; v)∈ I2 |C1(u; v)6 t})= (C1|C2)(t);

where the next to last step follows via the transformation u=F(x), v=G(y).

Thus the H2 distribution function of H1 is identical to the C2 distribution function of C1, and hence isinvariant under strictly increasing transformations of X and Y . As a consequence of Theorem 2.1, we shallrestrict ourselves to a discussion of distribution functions of copulas, and write 〈C1|C2〉 for 〈C1|C2〉(U; V )when the meaning is clear.

Example 2.1. The M; �; and W distribution functions of M; �; and W are readily evaluated from (2.1), andare presented in Fig. 1 (for t ∈ I):Thus, for example, 〈M |M 〉 is uniform on I; 〈M |�〉 has a beta distribution with parameters 1 and 2, 〈M |W 〉

is uniform on [0; 1=2], 〈�|M 〉 has a beta distribution with parameters 1=2 and 1, etc.

Example 2.2. Let U and V be uniform [0; 1] random variables whose joint distribution function is the copulaC. It is easy to verify that, for t in I, (M |C)(t)= 2t − �C(t), and thus the distribution functions of the orderstatistics of U and V can be expressed in terms of the C distribution of M :

df (min(U; V ))(t)= 2t − �C(t)= (M |C)(t);df (max(U; V ))(t)= �C(t)= 2t − (M |C)(t):

Also note that (C|M)(t)= �(−1)C (t), where �(−1)

C is the right-continuous “quasi-inverse” of �C , i.e., for t inI; �(−1)

C (t)= sup{u | �C(u)6 t}.

3. Orderings on the set of copulas

We now employ distribution functions of copulas to construct orderings on the set of copulas and hence,via Sklar’s theorem, on the set of bivariate distribution functions of continuous random variables.

De�nition 3.1. Let C, C1 and C2 be copulas. Then:

1. C1 is df -larger than C2 if 〈C1|C1〉¿st 〈C2|C2〉;2. C1 is C-larger than C2 if 〈C1|C〉¿st 〈C2|C〉.

280 R.B. Nelsen et al. / Statistics & Probability Letters 54 (2001) 277–282

The “df-larger” ordering was introduced by Cap)eraTa et al. (1997). A well-known partial order for copulasis concordance order: C1 is more concordant than C2, which we denote C1 C2, if C1¿C2 on I2. In thenext theorem, we characterize the concordance order by using the “C-larger” order for every copula C.

Theorem 3.1. Let C1 and C2 be copulas. Then C1 C2 if and only if C1 is C-larger than C2 for everycopula C.

Proof. First assume C1¿C2. Then {(u; v)∈ I2 |C1(u; v)6 t} ⊆ {(u; v)∈ I2 |C2(u; v)6 t} for all t in I; andhence �C({(u; v)∈ I2 |C1(u; v)6 t})6 �C({(u; v)∈ I2 |C2(u; v)6 t}) for every copula C. Thus (C1|C)6(C2|C), that is, 〈C1|C〉¿st 〈C2|C〉.Now assume there is a point (a; b) in (0; 1)2 such that t1 =C1(a; b)¡C2(a; b)= t2, and let t=(t1 + t2)=2.

We now construct a copula C with the property that (C1|C)(t)¿ (C2|C)(t), so that C1 is not C-larger than C2.Assume a6 b (the case a¿ b is similar). Let C be the copula whose probability mass is uniformly distributedon three line segments in I2; L1 connecting (0; 0) to ((t+a)=2; (t+a)=2), L2 connecting ((t+a)=2; (t−a+2b)=2)to (a−b+1; 1), and L3 connecting (a−b+1; (t−a+2b)=2) to (1; (t+a)=2) (C is a shu?e of M—see Mikusi)nskiet al. (1992)). Note that (a; b)∈L2. Let S1 = {(u; v)∈ I2 |C1(u; v)6 t} and S2 = {(u; v)∈ I2 |C2(u; v)6 t}.Then (a; b)∈ int(S1)∩ Sc

2 and �C(S2)¡�C([0; a]× [0; b])= a¡�C(S1), so that (C1|C)(t)¿ (C2|C)(t), whichcompletes the proof.

As a consequence of the above theorem, ordering C1 and C2 by requiring that C1 be C-larger than C2 forall C is too strong a requirement, since it is equivalent to concordance order. However, DeQnition 3:1:2 canbe used without the added “for all C” requirement to obtain new orderings. We illustrate with an example.

Example 3.1. The “M-larger” ordering. Let Ui and Vi be uniform [0; 1] random variables with copula Ci, fori=1; 2. Then

C1 is M -larger than C2 ⇔ 〈C1|M 〉¿st 〈C2|M 〉;⇔ (C1|M)6 (C2|M) ⇔ �(−1)

C16 �(−1)

C2⇔ �C1 ¿ �C2 ;

⇔ df (max(U1; V1))¿ df (max(U2; V2)) and df (min(U1; V1))6 df (min(U2; V2));

⇔min(U2; V2)6st min(U1; V1)6st max(U1; V1)6st max(U2; V2):

So, in a sense, C1 is M -larger than C2 if and only if the order statistics of U1 and V1 are stochastically“inside” the interval determined by the order statistics of U2 and V2.

4. Measures of association

Many measures of association are based on the notions of concordance and discordance. Recall that twoordered pairs (x1; y1) and (x2; y2) of real numbers are concordant if (x1− x2)(y1−y2)¿ 0; and discordant if(x1 − x2)(y1 − y2)¡ 0. The fundamental relationship between distribution functions of copulas and measuresof association which are based upon the probabilities of concordance and discordance is given in the nexttheorem.

Theorem 4.1. Let (X1; Y1) and (X2; Y2) be independent random vectors of continuous random variables withjoint distribution functions H1 and H2; respectively; with common marginal distribution functions F (of X1

and X2) and G (of Y1 and Y2); and let C1 and C2 denote the copulas of (X1; Y1) and (X2; Y2); respectively.

R.B. Nelsen et al. / Statistics & Probability Letters 54 (2001) 277–282 281

Let Q denote the diCerence between the probabilities of concordance and discordance of (X1; Y1) and(X2; Y2); i.e.;

Q=Pr[(X1 − X2)(Y1 − Y2)¿ 0]− Pr[(X1 − X2)(Y1 − Y2)¡ 0]:

Then Q is a function of C1 and C2; and is given by

Q=Q(C1; C2)= 3− 4∫ 1

0(C1|C2)(t) dt=3− 4

∫ 1

0(C2|C1)(t) dt:

Proof. Theorem 5:1:1 in Nelsen (1999) yields

Q=Q(C1; C2)= 4∫ 1

0

∫ 1

0C1(u; v) dC2(u; v)− 1=4

∫ 1

0

∫ 1

0C2(u; v) dC1(u; v)− 1;

or equivalently,

Q=4E(〈C1|C2〉(U2; V2))− 1=4E(〈C2|C1〉(U1; V1))− 1;

where Ui =F(Xi) and Vi =G(Yi); i=1; 2. But if T is a random variable with support in I and distributionfunction K(t), then E(T )= 1− ∫ 1

0 K(t) dt, from which the result follows.

Among the nonparametric measures of association which are related to concordance are the populationversions of Kendall’s &, Spearman’s ', Gini’s (, and Spearman’s footrule ’. Each of these coeMcients isa linear function of Q for various copulas (Nelsen, 1999). SpeciQcally, we have (A denotes the copula(M +W )=2):

Corollary 4.2. LetX andYbe continuous random variables with copula C; and let &C; 'C; (C; and ’C denotethe population versions ofKendall’s &; Spearman’s '; Gini’s (; and Spearman’s footrule ’; respectively.Then:1. &C =Q(C; C)= 3− 4

∫ 10 (C|C)(t) dt;

2. 'C =3Q(C;�)= 9− 12∫ 10 (C|�)(t) dt;

3. (C =2Q(C; A)= 6− 8∫ 10 (C|A)(t) dt; and

4. ’C = 32Q(C;M)− 1

2 = 4− 6∫ 10 (C|M)(t) dt.

The relationship between Kendall’s & and (C|C) is discussed by Cap)eraTa et al. (1997), where (C|C) isreferred to as a decomposition of Kendall’s & (also see Genest and Rivest, 1993). The analogous relationshipbetween Spearman’s ' and (C|�) is discussed by Garralda Guillem (1997). Other measures of associationcan be readily constructed from other distribution functions of copulas.Let C1; C2 be copulas, and let &1; &2; '1; '2; (1; (2; ’1; ’2 denote the corresponding values of Kendall’s &,

Spearman’s ', Gini’s (, and Spearman’s footrule ’, respectively. Then we have the following implicationsamong orderings based on distribution functions of copulas and measures of association, whose veriQcationsfollow from DeQnition 3.1 and Corollary 4.2:

Fig. 2. Implications among the various dependence orderings.

282 R.B. Nelsen et al. / Statistics & Probability Letters 54 (2001) 277–282

It is well known that the concordance order of C1 and C2 (C1 ≺ C2) implies the corresponding ordering ofthe four measures of association, i.e. &16 &2; '16 '2; (16 (2, and ’16’2. Note that the three “C-larger”orders (for C =�; A and M) occupy intermediate positions, in implication, between concordance order andorderings based on the values of the three corresponding measures of association ('; (; and ’); whereas the“df-larger” order, which is not comparable to concordance order, implies the order based upon values ofKendall’s &. Counterexamples exist to further implications among the orderings in Fig. 2 (see )Ubeda Flores,2001).

Acknowledgements

The authors wish to thank two anonymous referees for their helpful comments on an earlier version of thispaper.

References

Cap)eraTa, P., FougTeres, A.-L., Genest, C., 1997. A stochastic ordering based on a decomposition of Kendall’s tau. In: BeneXs, V., XStXep)an, J.(Eds.), Distributions with Given Marginals and Moment Problems. Kluwer Academic Publishers, Dordrecht, pp. 81–86.

Garralda Guillem, A.I., 1997. Dependencia y Tests de Rangos para Leyes Bidimensionales. Ph.D. Thesis, University of Granada, Spain.Genest, C., Rivest, L.-P., 1993. Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc. 88, 1034–1043.Mikusi)nski, P., Sherwood, H., Taylor, M.D., 1992. ShuYes of Min. Stochastica 13, 61–74.Nelsen, R.B., 1999. An Introduction to Copulas. Springer, New York.Sklar, A., 1959. Functions de r)epartition Ta n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231.)Ubeda Flores, M., 2001. C)opulas y Cuasic)opulas: Interrelaciones y Nuevas Propiedades; Aplicaciones. Ph.D. Thesis, University of Almer)/a,

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