Non-Parametric Estimation of the Limit Dependence Function of ...

Extremes 2:3, 245±268, 1999 # 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.

Non-Parametric Estimation of the Limit Dependence Function of Multivariate Extremes B. ABDOUS Dept. matheÂmatiques et informatique, Universite du QueÂbec aÁ Trois-RivieÁres, Trois-RivieÁres, QC, Canada G9A 5H7 E-mail: [email protected] K. GHOUDI Dept. matheÂmatiques et informatique, Universite du QueÂbec aÁ Trois-RivieÁres, Trois-RivieÁres, QC, Canada G9A 5H7 E-mail: [email protected] A. KHOUDRAJI Dept. matheÂmatiques, Universite Cadi Ayyad, Marrakech, Morocco [Received May 13, 1998; Revised August 30, 1999; Accepted November 16, 1999] Abstract. This paper presents a new estimation procedure for the limit distribution of the maximum of a multivariate random sample. This procedure relies on a new and simple relationship between the copula of the underlying multivariate distribution function and the dependence function of its maximum attractor. The obtained characterization is then used to de®ne a class of kernel-based estimates for the dependence function of the maximum attractor. The consistency and the asymptotic distribution of these estimates are considered. Key words. multivariate extremes, kernel estimation, dependence function, Gaussian process, regular variation AMS 1991 Subject Classi®cations.

1.

PrimaryÐ62G32; SecondaryÐ62G05.

Introduction

In many statistical applications the study of extreme values is of great importance. In hydrology, for example, the maximum ¯ow of a river is essential in the design of a dam. For pollution data, it is usually the highest level of pollutants that need be watched. Most of these studies reduce to the estimation of the joint distribution of the extremes of several characteristics. Though, results of this work are valid for multivariate extremes, for simplicity of the presentation, we only focus on the bivariate case. Sklar (1959) de®ned a copula or dependence function as any multivariate distribution function with uniform marginals. Given a bivariate random vector …X; Y† with distribution function H having continuous marginals H1 and H2 , there exists a unique copula C associated to H satisfying H…x; y† ˆ C…H1 …x†; H2 …y††, for any x and y. The dependence

246

ABDOUS, GHOUDI, AND KHOUDRAJI

structure between X and Y is perfectly characterized by C. Next, let …X1 ; Y1 †; . . . ; …Xn ; Yn † be a sequence of independent random vectors drawn from H… ? ; ? † and let …Xn ; Yn † ˆ …max…X1 ; . . . ; Xn †; max…Y1 ; . . . ; Yn ††: H is said to belong to the max-domain attraction of an extreme value distribution H with non degenerate marginals, if there exist sequences a1n 40, a2n 40, b1n and b2n such that lim P

n??

   Xn ÿ b1n Y  ÿ b2n  x; n y a1n a2n

ˆ lim H n …a1n x ‡ b1n ; a2n y ‡ b2n † n??

ˆ H  …x; y†:

…1†

Let C denote the copula associated to H  and let H1 and H2 be the marginals of H . From Galambos (1978), H belongs to the max-domain of attraction of H if and only if H1 and H2 belong to the domain of attraction of H1 and H2 respectively and if C satis®es C …x; y† ˆ lim C…x1=m ; y1=m †m ; m??

for all 0  x; y  1:

…2†

The marginals H1 and H2 are univariate extreme value distributions. Their characterization can be found in Galambos (1978). As for the copula C , Geoffroy (1958), Tiago de Olivera (1958), Sibuya (1960) and Pickands (1981) gave the following representation.    ln…y† ; for 0  x; y  1; C …x; y† ˆ exp ln…xy†A ln…xy†

…3†

where A is a convex function on [0,1] satisfying max…t; 1 ÿ t†  A…t†  1. The function A is often called the dependence function associated to C . Clearly, the estimation of H  may be accomplished via the estimation of its marginals and the dependence function A… ? †. In this work, we deal with A… ? † only, because the problem of estimating univariate extreme value distributions was considered by several authors (see, e.g. Davis and Resnick, 1984; Dekkers and de Haan, 1989). Note also that estimation of multivariate extreme-value distributions has been the subject of several papers, to name a few: Pickands (1981), Tiago de Oliveira, 1989, Tawn (1988), Smith, Tawn and Yuen (1990), Joe, Smith and Weissman (1992), Deheuvels (1991) and CapeÂraaÁ, FougeÁres and Genest (1997). Unfortunately, most of these authors assume that they dispose of a random sample from the distribution H  itself. In practice, such samples are hard to come by and if any, they usually suffer from the smallness of their sizes. To circumvent these dif®culties, de Haan and Resnick (1993) and Einmahl, de Haan and Sinha (1997) proposed estimates of H  based on a sample from H. However, their

MULTIVARIATE EXTREME DEPENDENCE FUNCTIONS

247

procedures need estimates of the normalizing sequences a1n ; a2n ; b1n and b2n in (1). Unlike all these previous works, we propose a non-parametric estimate of the dependence function A… ? † which does not require any auxiliary estimates and which is suitable for samples drawn from either H or H  . The following characterization constitutes a key element in our estimation procedure. From now on, we shall say that a copula C belongs to the domain of attraction of an extreme value copula C if and only if (2) holds. Theorem 1: Let C be an extreme value copula with associated dependence function A. Then a copula C belongs to the domain of attraction of C if and only if h i 1 ÿ C …1 ÿ u†1ÿt ; …1 ÿ u†t ˆ A…t†; Vt [ ‰0; 1Š: …4† lim u?0 u Remark: More generally, in the d dimensional case …d  2†, replaces …t; 1 ÿ t† by Pone d the vector …t1 ; . . . ; td †, where 0  tk  1 for k ˆ 1; . . . ; d and kˆ1 tk ˆ 1, and de®nes A from ‰0; 1Šd ÿ 1 into ‰0; 1Š. The key idea of the new estimation scheme consists in exploiting Theorem 1 to reduce the estimation of A… ? † to that of a quantity depending only on C the copula associated to the original data. To be speci®c let …U; V† be a random vector with distribution function C, and for each t [ ‰0; 1Š, let 8 1=…1 ÿ t† ; V 1=t g < 1 ÿ maxfU Wt ˆ 1 ÿ U : 1ÿV

for t [ …0; 1†; for t ˆ 0;

…5†

for t ˆ 1:

Observe that Wt takes values in [0,1] and its distribution function Gt satis®es 1ÿt t Gt …u† ˆ 1 ÿ C‰…1 ÿ u† ; …1 ÿ u† Š. Consequently, C belongs to, d…C †, the max domain of attraction of C if and only if lim

u?0

Gt …u† ˆ A…t†; u

Vt [ ‰0; 1Š:

…6†

From this equation one can view A…t† as a parameter of regular variation of the distribution function Gt at zero or as the value of its right-hand-side derivative at zero. In this paper, we shall adopt this last point of view and estimate A…t† by studying the local behavior of the distribution function Gt near zero. The next Section introduces a class of kernel-based non-parametric estimates of the dependence function A…t†. The presentation is divided into two subsections. The ®rst Subsection deals with the estimation procedure when the marginal distributions are assumed to be known. The second Subsection presents the case of unknown marginals.

248

ABDOUS, GHOUDI, AND KHOUDRAJI

Asymptotic behavior of these estimates is provided for each case. Practical issue on bandwidth selection and an example of application are given in Section 3. Proofs are provided in Section 4.

2.

Non-parametric estimates of A( ? )

Let …X1 ; Y1 †; . . . ; …Xn ; Yn † be a random sample drawn from a bivariate distribution function H with associated copula C and marginals H1 and H2 . Let Hn be the joint empirical distribution function and let H1n and H2n be the empirical marginal distribution functions. ÿ1 ÿ1 …u†; H2n …v†g. Denote by Cn the empirical copula function given by Cn …u; v† ˆ Hn fH1n ÿ1 where F … ? † stands for the left continuous inverse function of any distribution function F. Let also Cn …u; v† ˆ Hn fH1ÿ 1 …u†; H2ÿ 1 …v†g. By Theorem 1, the copula C is in d…C † if and only if the function 1ÿt t Gt …v† ˆ 1 ÿ Cf…1 ÿ v† ; …1 ÿ v† g has a linear behavior in the neighborhood of zero. Thus a natural estimate of A…t† can be obtained by ®tting locally a linear polynomial to the empirical estimate of Gt . To simplify the introduction of such estimator, it is useful to start by assuming that marginals are known. The more realistic and practical case of unknown marginal will be considered later.

2.1.

Known marginals

Assume that the marginals H1 and H2 are given. For each t; v [ ‰0; 1Š, we propose to use a weighted least squares criterion to ®t a local linear polynomial to the function Gt;n …v† ˆ 1 ÿ Cn f…1 ÿ v†1ÿt ; …1 ÿ v†t g. This is achieved by minimizing with respect to a the following criterion Z

1 0

 2 Kh …u ÿ v† Gt;n …u† ÿ au du;

…7†

where h (the smoothing parameter) controls the size of the neighborhood of v, K denotes a weight function (or kernel) and Kh …u† ˆ K…u=h†=h. The above integral is restricted to ‰0; 1Š, the support of Gt;n , in order to avoid the well-known boundary effects problem. The solution of (7), given by ^ v† ˆ A…t;

R1 0

uKh …u ÿ v†Gt;n …u† du ; R1 2 0 u Kh …u ÿ v† du

provides an estimate of the local behavior of the function Gt … ? † near v. Now, an estimate of A…t† is provided by

249

MULTIVARIATE EXTREME DEPENDENCE FUNCTIONS

^ ˆ A…t; ^ 0† ˆ A…t† R1 0

ˆ

R1 0

uKh …u†Gt;n …u† du R1 2 0 u Kh …u† du

uKh …u†…1 ÿ Cn f…1 ÿ u†1 ÿ t ; …1 ÿ u†t g† du : R1 2 0 u Kh …u† du

…8†

Remark: As noted earlier, A…t† can be viewed as a parameter of regular variation of the function Gt at zero. Indeed, recall that a special class of regularly varying distribution functions of order ÿ a at zero, is given by the set of distributions F satisfying F…0† ˆ 0 and for which there exists a constant b such that limx?0 F…x†=xa ˆ b. In fact from (6), one sees that A(t) plays the role of the constant b in the regularly varying distribution function Gt at zero. Here the index of regular variation a is equal to 1. Estimation of the parameters of regularly varying distribution functions received considerable attention in the literature, see for instance Hill (1975), Hall (1982), CsoÈrgoÈ, Deheuvels and Mason (1985), Haeusler and Teugels (1985), Dekkers and de Haan (1989), and Beirlant, Vynckier and Teugels (1996) and the references therein. Most of these references focus on the estimation of the parameter a, however, Hill (1975) and Hall (1982) proposed an estimate of the parameter b, which in our context becomes A^H …t† ˆ

r ; nWt;…r†

…9†

where 1  r  n ÿ 1 and Wt;…r† is the rth order statistic of the sequence h i Wt;i ˆ 1 ÿ max H1 …Xi †1=…1 ÿ t† ; H2 …Yi †1=t ;

i ˆ 1; . . . ; n:

This estimate is in fact a special case of (8) obtained by choosing K as the Dirac mass function at zero and taking v ˆ Wt;…r† . The following notation will be used in the sequel R1

uKh …u†Gt …u† du R1 2 0 u Kh …u† du   R1 1ÿt t uK …u† 1 ÿ Cf…1 ÿ u† ; …1 ÿ u† g du h 0 : ˆ R1 2 0 u Kh …u† du

m…t; h† ˆ

0

…10†

^ ? †. The next The rest of this Subsection is devoted to the asymptotic behavior of A… Theorem states the consistency results. Theorem 2: If K is a bounded positive function satisfying Z 05 v2 K…v† dv5?:

250

ABDOUS, GHOUDI, AND KHOUDRAJI

If C is in the max-domain of attraction of C and h ˆ hn goes to 0 as n tends to in®nity, then the following hold. ^ ÿ A…t†j ˆ 0 in probability. (i) If limn?? nh2 ˆ ?, then limn?? sup0  t  1 jA…t† (ii) If ? X

exp…ÿ gnh2 †5?;

for every g40;

…11†

nˆ1

^ ÿ A…t†j ˆ 0 almost surely. then, lim sup jA…t† n?? 0  t  1

Remark: The conditions on h are not optimal and are used for simplicity of the proof. The optimal conditions can be obtained using the results on the rate of growth of the weighted multivariate empirical process (see for instance Alexander (1987)) instead of the rate of growth of the empirical process. Unfortunately, this will lengthen the proof considerably. Since t lives in ‰0; 1Š, one gets, as an immediate consequence, the following Lp convergence results. Corollary 1: Under the assumption (i) and the assumption (ii) of Theorem 2, Z lim

n??

1

0

^ ÿ A…t†jp dt jA…t†

1=p ˆ 0;

in probability and almost surely respectively, for any p  1. ^ ? † is established in the next Theorem where it is shown that The weak convergence of A… the process an …t† ˆ

p ^ ÿ m…t; h† nh A…t†

…12†

converges weakly to a mean zero Gaussian process. Theorem 3: sup

If K is a bounded positive function satisfying

ÿ ?5v5?

v4 K…v†5? and

Z 05

jvj3 K…v†dv5?:

If C is in the max-domain of attraction of C and lim n?? …hn ‡ …nhn †

…13† ÿ1

† ˆ 0, then an …t†

MULTIVARIATE EXTREME DEPENDENCE FUNCTIONS

251

converges weakly to a continuous Gaussian process with covariance function G… ? ; ? † satisfying R? R? G…s; t† ˆ

0

0

xyK…x†K…y†Y…x; y; s; t†dx dy ; ÿR ?
2 2 0 x K…x†dx

where  Y…x; y; s; t† ˆ

xA…t† ‡ yA…s†     …txVsy† ÿ ……1 ÿ t†xV…1 ÿ s†y† ‡ …txVsy† A ……1 ÿ t†xV…1 ÿ s†y† ‡ …txVsy†

with xVy denoting the maximum of x and y. This result could be used to construct con®dence regions for A… ? †. Pointwise con®dence intervals are also provided by using the above Theorem and the following estimate of the ^ variance of A…t† ^ ^ t† ˆ A…t† G…t;

R? R? 0

0

xy min…x; y†K…x†K…y†dx dy ; ÿR ?
2 2 0 x K…x†dx

^ in the expression of G…t; t†. which is obtained by substituting A…t† by A…t† Remark: In general, the estimate A^ is not convex and does not respect ^  1. This problem is not special to the estimator presented here, max…t; 1 ÿ t†  A…t† but it is common to most of the estimator of A found in the literature, see for example Deheuvels (1991). A possible solution to bypass this drawback is achieved by modifying (7) as follows Z Minimize

1 0

 2 Kh …u ÿ v† Gt;n …u† ÿ a…t†u du

Subject to: a…t† is convex max…t; 1 ÿ t†  a…t†  1; for t [ ‰0; 1Š: Smoothing under constraints has been successfully considered by many authors, see for instance Delecroix, Simioni and Thomas-Agnan (1996) and Mammen and ThomasAgnan (1999). We intend to tackle the minimization problem above in a forthcoming paper. Theorem 3, still holds if one replaces m…t; h† by A…t† in (12) provided p nh…A…t† ÿ m…t; h†† converges to zero as n goes to in®nity. This is achieved by conditions on the smoothness of the copula function C and on the speed of convergence of h to zero.

252 2.2.

ABDOUS, GHOUDI, AND KHOUDRAJI

Unknown marginals

If the marginals are unknown, then the same procedure can be repeated with Cn and Gt;n …v† 1ÿt t replaced by Cn and Gt;n …v† ˆ 1 ÿ Cn f…1 ÿ v† ; …1 ÿ v† g respectively. This yields the following class of estimates. ~ v† ˆ A…t;

R1 0

uKh …u ÿ v†Gt;n …u†du : R1 2 0 u Kh …u ÿ v†du

~ ˆ A…t; ~ 0†. Again the estimate of A…t† is given by A…t†

Remark: The analogous of (9) is given by A~H …t† ˆ

r ; ~ nWt;…r†

~t;…r† is the rth order statistic of the sequence where 1  r  n ÿ 1 and W h i ~t;i ˆ 1 ÿ max H1n …Xi †1=…1 ÿ t† ; H2n …Yi †1=t ; W

i ˆ 1; . . . ; n:

~ ? † are summarized in the next Theorem. To simplify its The asymptotic properties of A… statement, set C…x; y; s; t† ˆ 3fxA…t† ‡ yA…s† ‡ x ‡ yg ÿ f…1 ÿ t†xV…1 ÿ s†yg ÿ …txVsy†   tx ÿ ‰f…1 ÿ t†xV…1 ÿ s†yg ‡ txŠA f…1 ÿ t†xV…1 ÿ s†yg ‡ tx   …txVsy† ÿ ‰…1 ÿ t†x ‡ …txVsy†ŠA …1 ÿ t†x ‡ …txVsy†   sy ÿ ‰f…1 ÿ t†xV…1 ÿ s†yg ‡ syŠA f…1 ÿ t†xV…1 ÿ s†yg ‡ sy   …txVsy† ÿ ‰…1 ÿ s†y ‡ …txVsy†ŠA …1 ÿ s†y ‡ …txVsy†   sy ÿ ‰…1 ÿ t†x ‡ syŠA …1 ÿ t†x ‡ sy

253

MULTIVARIATE EXTREME DEPENDENCE FUNCTIONS



 tx ÿ ‰tx ‡ …1 ÿ s†yŠA tx ‡ …1 ÿ s†y ÿ ‰f…1 ÿ t†xV…1 ÿ s†yg ‡ …txVsy†ŠA



 …txVsy† ; f…1 ÿ t†xV…1 ÿ s†yg ‡ …txVsy†

for 0  x; y; s; t  1. Theorem 4: Let K be a bounded positive function satisfying (13). Assume that C is in the max-domain of attraction of C and that it admits continuous ®rst order partial derivatives. If h ˆ hn goes to 0 as n tends to in®nity, then the following hold. ~ ÿ A…t†j converges to zero in probability. (i) If limn!? nh2 ˆ ?, then sup0t1 jA…t† ~ ÿ A…t†j ˆ 0 almost surely. (ii) If (11) is satis®ed then limn!? sup0t1 jA…t† p 2 2 ~ ÿ m…t; h† converges in (iii) If limn!? nh =……ln n† …ln ln n†† ˆ ?, then nh A…t† distribution to a continuous Gaussian process with covariance function given by R? R? xyK…x†K…y†C…x; y; s; t†dx dy : G…s; t† ˆ 0 0 ÿR ?
2 2 0 x K…x†dx

3.

Selection of the bandwidth h

We give in this Section some guidelines to select the smoothing parameter h. We deliberately omit details because we will present them together with a thorough simulation study in a forthcoming paper. Since our estimation problem reduces essentially to a local density estimation problem, we can adopt any of the many existing local bandwidth selectors (see Hazelton (1996) for a ~ and review). To see how this can be achieved, recall that for each ®xed t [ ‰0; 1Š, both A…t† ^ A…t† estimate the probability density of the r.v. Wt , de®ned by (5), at the support's endpoint 0. Thus, a natural procedure of selecting the bandwidth h should be based on a local criterion. The most popular and tractable measure of local error is the Mean Squared Error (MSE), namely  ÿ A…t†Š2 ; M…h† ˆ E‰A…t†  stands for either A…t† ~ or A…t†. ^ where A…t† Since A… ? † is unknown, the optimal parameter  h ˆ argminh M…h† cannot be evaluated in practice. One has to use Taylor series expansion of M…h† and then estimate the unknowns by means of plug-in or cross-validation techniques. Also, instead of appealing to asymptotic approximations of M…h†, we can

254

ABDOUS, GHOUDI, AND KHOUDRAJI

Figure 1.

estimate the criterion M…h† by means of the smoothed local cross-validation or bootstrap methods. Finally, note that these selectors will deliver a bandwidth which varies with t. To reduce computations, we can consider minimizing a global criterion. To this end, we propose to integrate M…h† with respect to t, i.e. we seek for a global bandwidth h which minimizes Z IM…h† ˆ

1 0

 ÿ A…t†Š dt: E‰A…t† 2

Here again, we will face the problem of estimating IM…h†. But this dif®culty can be easily bypassed by adopting one of the various global bandwidth selectors, see, e.g. Devroye (1997) for a complete and recent review. In the following simulation study, the L1 -double kernel method (Devroye (1997)) has been used to select the bandwidths. The following ®gures show the graphs of the theoretical curve of dependence function A, Hill's estimate and the estimate Aà discussed in this paper with h selected using the double kernel method. Figure 1 (a) shows the result for a sample of 500 observations of independent uniform random vectors. For Figure 1 (b), a sample of size 500 was generated from the Gumbel-Hougaard family (Gumbel (1960)) with parameter y ˆ 2. The optimal choice for h was 0.9346 for the ®rst simulation and 0.3316 for the second.

4.

Proofs

Before dealing with proofs of the results stated earlier, we begin by considering the next Lemma which will be used repeatedly.

MULTIVARIATE EXTREME DEPENDENCE FUNCTIONS

255

Lemma 1: Let C be an extreme value copula with associated dependence function A. Then a copula C belongs to the domain of attraction of C if and only if 1 ÿ C…1 ÿ u; 1 ÿ v† ˆ A…t† u;v?0 u‡v lim

…14†

holds for any sequence …u; v† in …0; 1†2 satisfying limu;v?0 v=…u ‡ v† ˆ t. Proof of Lemma 1: The proof is achieved by showing that (14) is equivalent to (2). First, let us show that (14) implies (2). Clearly, (2) holds for x ˆ 0; 1 or y ˆ 0; 1. Next, for each 05x; y51 and any integer m40, let u ˆ 1 ÿ x1=m and v ˆ 1 ÿ y1=m . Note that as m goes to in®nity u and v go to zero and v=…u ‡ v† goes to t ˆ ln…y†= ln…xy†. Now, observe that h i lnC…x1=m ; y1=m † 1 ÿ C…x1=m ; y1=m † m…2 ÿ x1=m ÿ y1=m † 1=m 1=m ln…xy†: m ln C…x ; y † ˆ ln…xy† 1 ÿ C…x1=m ; y1=m † 2 ÿ x1=m ÿ y1=m The ®rst and the last fractions go to ÿ 1 as m goes to in®nity, since limz?1 …1 ÿ z†= ln…z† ˆ ÿ 1 and limz?0 ‰1 ÿ exp…z†Š=z ˆ ÿ1. By applying (14), one sees that the second fraction converges to Afln…y†= ln…xy†g. Taking the exponential of both sides yields (2). To complete the proof it remains to show that (2) implies (14). Assume that (2) holds, then for any sequence …um ; vm † ˆ …1 ÿ x1=m ; 1 ÿ y1=m †, where 05x; y51 and m is an arbitrary integer, one has 1 ÿ C…1 ÿ um ; 1 ÿ vm † 1 ÿ C…1 ÿ um ; 1 ÿ vm † 1 m lnfC…1 ÿ um ; 1 ÿ vm †g: ˆ um ‡ vm lnfC…1 ÿ um ; 1 ÿ vm †g m…um ‡ vm † Quite similar arguments to those used previously give lim

um ;vm ?0

1 ÿ C…1 ÿ um ; 1 ÿ vm † ˆ A…t†; um ‡ v m

…15†

where t ˆ limm?? vm =…um ‡ vm †. Next, it will be shown that for any a40 and 0  u; au  1 one has   1 ÿ C…1 ÿ u; 1 ÿ au† a ˆA : u?0 …1 ‡ a†u a‡1 lim

…16†

For, observe that 0

1 a ÿ  1; ln…1 ÿ u† ln…1 ÿ au†

…17†

256

ABDOUS, GHOUDI, AND KHOUDRAJI

for 0  a  1 and 0  u  1 and ÿa 

1 a ÿ  0; ln…1 ÿ u† ln…1 ÿ au†

…18†

for a  1 and 0  au  1. Let x ˆ exp…ÿ 1† and let y ˆ xa then for any 05u; au51 there exists an integer m such that um ‡ k  u  um and vm ‡ k  au  vm , where um ˆ 1 ÿ x1=m , vm ˆ 1 ÿ y1=m and k is the integer part of a ‡ 2. To see this it suf®ces to show that there exists an m such that ÿ1 ÿ1 ÿk m ln…1 ÿ u† ln…1 ÿ u† and ÿa ÿa ÿk m : ln…1 ÿ au† ln…1 ÿ au† That is 

   ÿ1 ÿa ÿ1 ÿa ; ; ÿ k  m  min : max ln…1 ÿ u† ln…1 ÿ au† ln…1 ÿ u† ln…1 ÿ au† Such m exists if ÿ1 ÿ a ÿ  1: kÿ ln…1 ÿ u† ln…1 ÿ au† By (17) and (18), the above holds if k ÿ max…a; 1†  1, which is true by the choice of k. Note that such m goes to in®nity as u goes to zero. Moreover, limm?? vm =…um ‡ vm † ˆ a=…a ‡ 1† and limm?? …um‡k ‡ vm‡k †=…um ‡ vm † ˆ 1. To complete the proof one uses the monotonicity of C to obtain 1 ÿ C…1 ÿ um ‡ k ; 1 ÿ vm ‡ k † um ‡ k ‡ vm ‡ k 1 ÿ C…1 ÿ u; 1 ÿ v†  um ‡ v m um ‡ k ‡ v m ‡ k 1 ‡ au 

1 ÿ C…1 ÿ um ; 1 ÿ vm † um ‡ vm ; um ‡ k ‡ vm ‡ k um ‡ v m

taking the limit as u goes to zero (i.e. m going to in®nity) yields the result. Next, recall the following property of copula function jC…x1 ; y1 † ÿ C…x2 ; y2 †j



jx1 ÿ x2 j ‡ jy1 ÿ y2 j;

…19†

257

MULTIVARIATE EXTREME DEPENDENCE FUNCTIONS

for 0  x1 ; x2 ; y1 ; y2  1. To complete the proof of the Lemma, let …u; v† be a sequence such that limu;v?0 v=…u ‡ v† ˆ t. If t ˆ 1, then (19) yields 1 ÿ C…1 ÿ u; 1 ÿ v† ÿ 1 lim u;v?0 u‡v



2 lim

u;v?0 u

u ˆ 0: ‡v

Whereas, when t51 then by letting a ˆ t=…1 ÿ t† one gets 1 ÿ C…1 ÿ u; 1 ÿ v† 1 ÿ C…1 ÿ u; 1 ÿ au† …1 ‡ a†u ˆ u‡v …1 ‡ a†u u‡v ‡

C…1 ÿ u; 1 ÿ au† ÿ C…1 ÿ u; 1 ÿ v† : u‡v

Now (14) follows from (19) and (16).

&

Proof of Theorem 1: From Lemma 1, it suf®ces to show that (4) is equivalent to (14). Clearly, (14) entails (4). Conversely, assume that (4) holds. Consider any sequence …u; v† such that limu;v?0 v=…u ‡ v† ˆ t. The case t ˆ 1 is treated in the proof of Lemma 1, thus for t51, one has 1ÿt

1 ÿ C…1 ÿ u; 1 ÿ v† 1 ÿ C……1 ÿ x† ˆ u‡v x ‡

t

; …1 ÿ x† †

x u‡v

C……1 ÿ x†1 ÿ t ; …1 ÿ x†t † ÿ C…1 ÿ u; 1 ÿ v† ; u‡v

1=…1 ÿ t†

where x ˆ 1 ÿ …1 ÿ u† . As u goes to zero, the ®rst fraction on the right-hand-side converges to A(t) by (4), the second fraction converges to 1 by de®nition of x and t and from (19), the last fraction is bounded by j…1 ÿ u†t=…1 ÿ t† ÿ 1 ‡ vj=…u ‡ v†, which goes to zero. & Proof of Theorem 2:

Use notation (10) to write

^ ÿ A…t†j  sup jm…t; h† ÿ A…t†j ‡ sup jA…t† ^ ÿ m…t; h†j: sup jA…t†

0t1

0t1

0t1

The ®rst term on the right-hand-side of the above inequality can be bounded as follows R? jm…t; h† ÿ A…t†j 

0

IIf05v51=hgjf…t; hv†jv2 K…v† dv R? ; 2 0 IIf05v51=hgv K…v† dv

258

ABDOUS, GHOUDI, AND KHOUDRAJI

where II… ? † denotes the indicator function and ( f…t; u† ˆ

Gt …u† u

0

ÿ A…t†; for 0  t  1 and 05u  1 for 0  t  1 and u ˆ 0.

In order to use the dominated convergence Theorem, we have to bound f… ? ; ? †. To this end, note that for any copula function one has max…0; u ‡ v ÿ 1†  C…u; v†  min…u; v†; for 0  u; v  1:

…20†

One also has …1 ÿ u†t  1 ÿ tu;

for 0  u; t  1;

…21†

…1 ÿ u†t  1 ÿ 2tu;

for 0  t  1 and 0  u  1=2:

…22†

Now, by using (20)±(21) and the well-known fact max…t; 1 ÿ t†  A…t†  1;

for all t [ ‰0; 1Š;

one gets max…t; 1 ÿ t† ÿ 1  f…t; u†;

for all 0  t; u  1:

…23†

Furthermore, since C lies between 0 and 1, then when u41=2 f…t; u†  2 ÿ max…t; 1 ÿ t†;

for all 0  t  1:

…24†

This inequality is still valid when u  1=2 by virtue of (20) and (22). Finally, by combining (23) and (24) we end up with jf…t; u†j  3=2;

for all t [ ‰0; 1Š:

Next, since C and A are continuous and C is in d…C †, then f…t; u† is continuous on ‰0; 1Š2. Consequently, C is uniformly continuous. The uniform continuity and the fact that f…t; 0† ˆ 0 for all t in ‰0; 1Š imply that sup0  t  1 jf…t; u†j converges to zero as u goes to zero. Upon noting that by hypothesis u2 K…u† is integrable, and calling on the dominated convergence Theorem twice, one obtains R? lim sup jm…t; h† ÿ A…t†j 

n?? 0  t  1

0

lim sup jf…t; hv†jv2 K…v† dv

h?0 0  t  1 R? 2 0 v K…v† dv

ˆ 0:

259

MULTIVARIATE EXTREME DEPENDENCE FUNCTIONS

^ ÿ m…t; h†j. Observe that It remains to study sup0  t  1 jA…t†

^ ÿ m…t; h†j  h sup jA…t†

0t1

ÿ1

sup

R 1h

0  u;v  1

jCn …u; v† ÿ C…u; v†j R 10

uK…u† du

2 0 u K…u† du h

;

and that R 1h 0

uK…u† du

0

u2 K…u† du

R 1h

R?

0 ÿ? R ? 0

uK…u† du u2 K…u† du

;

as n??:

^ ÿ m…t; h†j Hence, by applying Theorem 1 of Kiefer (1961), we see that sup0  t  1 jA…t† 2 converges to zero in probability if nh goes to in®nity, and it converges to zero almost P 2 exp…ÿ gnh † is ®nite. & surely if for every g40 the ? nˆ1 Next, before dealing with Theorem 3, we need to consider the following technical Lemma. Lemma 2: Z

1=h 0

If K is a bounded positive function satisfying (13) then for any a; b40 a

…1 ÿ …1 ÿ hu† †uK…u† du  c1 ah;

…25†

…1 ÿ …1 ÿ hu†a †…1 ÿ …1 ÿ hu†b †uK…u† du  c2 abh2 ;

…26†

and Z

1=h 0

where c1 and c2 are positive constants depending only on K. Proof of Lemma 2: a

The proof of this Lemma rests on the following inequality,

1 ÿ …1 ÿ u†  a maxf2u; ÿ ln…1 ÿ u†= ln…2†g; for all a40 and 0  u  1:

…27†

260

ABDOUS, GHOUDI, AND KHOUDRAJI

Note that Z

1=h

0

a

…1 ÿ …1 ÿ hu† †uK…u†du Z

 2ah

1=2h

0

Z  2ah

? 0

Z  2ah

? 0

Z

a u K…u†du ÿ ln…2† 2

4ah2 u K…u†du ÿ ln…2† 2

1=h

1=2h

Z

1=h

1=2h

ln…1 ÿ hu†uK…u†du

ln…1 ÿ hu†u3 K…u†du

4ah 3 sup…juj K…u†† u K…u†du ÿ ln…2† u40 2

Z

1

1=2

ln…1 ÿ u†du:

This concludes the proof of (25) since supu40 …juj3 K…u†† is ®nite by (13). As for (26), similar arguments give Z

1=h 0

…1 ÿ …1 ÿ hu†a †…1 ÿ …1 ÿ hu†b †uK…u†du

 4abh2

Z

? 0

u3 K…u†du ‡

8abh2 ln…2†

sup…u4 K…u†† 2 u40

Z

1

1=2

ln…1 ÿ u†2 du;

which implies (26). Proof of Theorem 3:

& In the sequel, we shall denote

 …u; t† ˆ pn…C ……1 ÿ u†1 ÿ t ; …1 ÿ u†t † ÿ C……1 ÿ u†1 ÿ t ; …1 ÿ u†t ††; b n n and Z Djh ˆ

0

1=h

uj K…u†du;

for 1  j  3:

Use these notations to rewrite (12) as follows 1 an …t† ˆ ÿ p hD2h

Z 0

1=h

 …hu; t†du: uK…u†b n

The weak convergence of an … ? †, is obtained by establishing its tightness and the convergence of its ®nite dimensional distributions. The tightness of an … ? † is proved in three steps. Namely,

261

MULTIVARIATE EXTREME DEPENDENCE FUNCTIONS

Step 1. There exists k40 such that for all 1=nh  t1 5t5t2  1 ÿ 1=nh R…t1 ; t; t2 † ˆ E‰fan …t† ÿ an …t1 †g2 fan …t2 † ÿ an …t†g2 Š  k…t2 ÿ t1 †2 : Step 2. lim

sup

n?? 0  s 5 t  1=nh

jan …t† ÿ an …s†j ˆ 0;

in probability:

Step 3. lim

sup

n?? 1ÿ1=nh  s 5 t  1

jan …t† ÿ an …s†j ˆ 0;

in probability:

Hereafter, we outline the proof of steps 1 and 2, we omit the proof of Step 3 because it is similar to that of Step 2. First consider Step 1. To simplify the expression of R…t1 ; t; t2 † set 1ÿt

zi …u; t† ˆ IIfUi  …1 ÿ u†

t

; Vi  …1 ÿ u† g ÿ Cf…1 ÿ u†

1ÿt

t

; …1 ÿ u† g;

and Zi …u; s; t† ˆ zi …u; t† ÿ zi …u; s†: Then, by using jZi …u; s; t†j  2; it follows that 1 R…t1 ; t; t2 † ˆ 2 4 h D2h

Z

4 Y

…0;1=h†4 i ˆ 1

ui K…ui †

n 1 X E‰Zk1 …hu1 ; t1 ; t† n2 k ;...;k ˆ 1 1

4

6Zk2 …hu2 ; t1 ; t†Zk3 …hu3 ; t; t2 †Zk4 …hu4 ; t; t2 †Šdu1 du2 du3 du4  R1 …t1 ; t; t2 † ‡ R2 …t1 ; t; t2 †; where 4D2 R1 …t1 ; t; t2 † ˆ 2 1h4 nh D2h

Z …0;1=h†2

EjZ1 …hu1 ; t1 ; t†Z1 …hu2 ; t; t2 †j

2 Y iˆ1

ui K…ui †dui ;

and R2 …t1 ; t; t2 † ˆ

12D21h h2 D42h

Z …0;1=h†2

EjZ1 …hu1 ; t1 ; t†jEjZ2 …hu2 ; t; t2 †j

2 Y iˆ1

ui K…ui †dui :

262

ABDOUS, GHOUDI, AND KHOUDRAJI

An upper-bound of R2 …t1 ; t; t2 † is given by 48D21h h2 D42h

Z …0;1=h†2

fP…A1 † ‡ P…B1 †gfP…A2 † ‡ P…B2 †g

2 Y iˆ1

ui K…ui †dui ;

where A1 ˆ f…x; y† : …1 ÿ hu1 †1 ÿ t1  x  …1 ÿ hu1 †1 ÿ t and 0  y  …1 ÿ hu1 †t g; B1 ˆ f…x; y† : 0  x  …1 ÿ hu1 †1 ÿ t1 and …1 ÿ hu1 †t  y  …1 ÿ hu1 †t1 g; A2 ˆ f…x; y† : …1 ÿ hu2 †1 ÿ t  x  …1 ÿ hu2 †1 ÿ t2 and 0  y  …1 ÿ hu2 †t2 g; B2 ˆ f…x; y† : 0  x  …1 ÿ hu2 †

1ÿt

t

t

and …1 ÿ hu2 † 2  y  …1 ÿ hu2 † g:

For i ˆ 1; 2, an application of (19) yields ÿ
maxfP…Ai †; P…Bi †g  1 ÿ …1 ÿ hui †t2 ÿ t1 :

…28†

Consequently, by (25), there exists a constant k0 such that R2 …t1 ; t; t2 †  k0

D21h 2 …t ÿ t1 † : D42h 2

Next, straightforward computations show that the expectation involved in R1 …t1 ; t; t2 † is less than P…A1 \ A2 † ‡ P…A1 \ B2 † ‡ P…B1 \ A2 † ‡ P…B1 \ B2 † ‡ 12…P…A1 † ‡ P…B1 ††…P…A2 † ‡ P…B2 ††: Note that …A1 \ A2 † is not empty only if u1  u2  …1 ÿ …1 ÿ hu1 †…1 ÿ t1 †=…1 ÿ t2 † †=h. Therefore, by (28), ÿ

P…A1 \ A2 †  1 ÿ …1 ÿ hu1 †

t2 ÿ t1


8