Large-sample tests of extreme-value dependence for multivariate copulas

# Large-sample tests of extreme-value dependence for multivariate copulas

Laboratoire de mathématiques et applications, UMR CNRS 5142
Université de Pau et des Pays de l’Adour
B.P. 1155, 64013 Pau Cedex, France
Johan Segers
Institut de statistique, biostatistique et sciences actuarielles
Université catholique de Louvain
Voie du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium
johan.segers@uclouvain.be
Jun Yan
Department of Statistics
University of Connecticut, 215 Glenbrook Rd. U-4120
Storrs, CT 06269, USA
jun.yan@uconn.edu
###### Abstract

Starting from the characterization of extreme-value copulas based on max-stability, large-sample tests of extreme-value dependence for multivariate copulas are studied. The two key ingredients of the proposed tests are the empirical copula of the data and a multiplier technique for obtaining approximate -values for the derived statistics. The asymptotic validity of the multiplier approach is established, and the finite-sample performance of a large number of candidate test statistics is studied through extensive Monte Carlo experiments for data sets of dimension two to five. In the bivariate case, the rejection rates of the best versions of the tests are compared with those of the test of GhoKhoRiv98 recently revisited by BenGenNes09. The proposed procedures are illustrated on bivariate financial data and trivariate geological data.

Keywords: max-stability, multiplier central limit theorem, pseudo-observations, ranks.

## 1 Introduction

Let be a -dimensional random vector with continuous marginal cumulative distribution functions (c.d.f.s) . It is then well-known from the work of Skl59 that the c.d.f.  of can be written in a unique way as

 F(x)=C{F1(x1),…,Fd(xd)},x∈Rd,

where the function is a copula and can be regarded as capturing the dependence between the components of . If, additionally, is max-stable, i.e., if

 C(u)={C(u1/r1,…,u1/rd)}r,∀u∈[0,1]d,∀r>0, (1)

the function is an extreme-value copula. Such copulas arise in the limiting joint distributions of suitably normalized componentwise maxima (Gal87; GudSeg10) and are the subject of increasing practical interest in finance (McNFreEmb05), insurance (FreVal98) and hydrology (SalDeMKotRos07).

Given a random sample from c.d.f. , it is of interest in many applications to test whether the unknown copula belongs to the class of extreme-value copulas. A first solution to this problem was proposed in the bivariate case by GhoKhoRiv98 who derived a test based on the bivariate probability integral transformation. The suggested approach was recently improved by BenGenNes09 who investigated the finite-sample performance of three versions of the test.

The aim of this paper is to study tests of extreme-value dependence for multivariate copulas based on characterization (1). The first key element of the proposed approach is the empirical copula of the data which is a nonparametric estimator of the true unknown copula. Starting from characterization (1), the empirical copula can be used to derive natural classes of empirical processes for testing max-stability. As the distribution of these processes is unwieldy, one has to resort to a multiplier technique to compute approximate -values for candidate test statistics. This is the second key element of the proposed approach and is based on the seminal work of Sca05 and RemSca09, revisited recently in Seg11. The outcome of this work is a general procedure for testing extreme-value dependence which, in principle, can be used in any dimension.

The second section of the paper is devoted to recent results on the weak convergence of the empirical copula process obtained in Seg11. A detailed and rigorous description of the proposed tests is given in Section 3, while their implementation is discussed in Section 4. In the fifth section, the results of an extensive Monte Carlo study are partially reported. They are used to provide recommendations in Section 6 enabling the proposed approach to be safely used to test extreme-value dependence in data sets of dimension two to five. The test based on one of the best performing statistics is finally used to test bivariate extreme-value dependence in the well-known insurance data of FreVal98, and trivariate extreme-value dependence in the uranium exploration data of CooJoh86.

The following notational conventions are adopted in the sequel. The arrow ‘’ denotes weak convergence in the sense of Definition 1.3.3 in vanWel96, and represents the space of all bounded real-valued functions on equipped with the uniform metric. Also, for any and any , we adopt the notation . Furthermore, the set of extreme-value copulas, i.e., copulas satisfying (1), is denoted by .

Note finally that all the tests studied in this work are implemented in the R package copula (KojYan10) available on the Comprehensive R Archive Network (Rsystem).

## 2 Weak convergence of the empirical copula process

Let , , be pseudo-observations from the copula computed from the data by , where is the rank of among . The pseudo-observations can equivalently be rewritten as , where is the empirical c.d.f. computed from , and where the scaling factor is classically introduced to avoid problems at the boundary of . The proposed tests are based on the empirical copula of the data (Deh79; Deh81), which is usually defined as the empirical c.d.f. computed from the pseudo-observations, i.e.,

 Cn(u)=1nn∑i=11(^Ui≤u),u∈[0,1]d.

For any , let be the partial derivative of with respect to its th argument at , i.e.,

 C[j](u)=limh→0uj+h∈[0,1]C(u1,…,uj−1,uj+h,uj+1,…,ud)−C(u)h,u∈[0,1]d.

It is well-known (see e.g., Nel06, Theorem 2.2.7) that exists almost everywhere on and that, for those for which it exists, . If exists and is continuous on for all , then, from Corollary 5.3 of vanWel07 (see also Stu84; GanStu87; FerRadWeg04; Tsu05), the empirical copula process converges weakly in to the tight centered Gaussian process

 C(u)=α(u)−d∑j=1C[j](u)α(1,…,1,uj,1,…,1),u∈[0,1]d, (2)

where is a -Brownian bridge, i.e., a tight centered Gaussian process on with covariance function , . Without loss of generality, we assume in the sequel that has continuous sample paths.

For many copula families however, the partial derivatives , , fail to be continuous on the whole of . For instance, as shown in Seg11, many popular bivariate extreme-value copulas have discontinuous partial derivatives at and . To deal with such situations, Seg11 considered the following less restrictive condition:

()

for any , exists and is continuous on the set .

Under Condition (), for any , Seg11 extended the domain of to the whole of by setting

 C[j](u)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩limsuph↓0C(u1,…,uj−1,h,uj+1,…,ud)h,if u∈[0,1]d,uj=0,limsuph↓0C(u)−C(u1,…,uj−1,1−h,uj+1,…,ud)h,if u∈[0,1]d,uj=1,

which ensures that the process defined in (2) is well-defined on the whole of , and showed the weak convergence of the empirical copula process to in . Condition () was verified in Seg11 for many popular families including many -dimensional extreme-value copulas.

## 3 Description of the tests

Starting from characterization (1) and having at hand a nonparametric copula estimator such as , it seems natural to base tests of the hypothesis on processes of the form

 Dr,n(u)=√n[{Cn(u1/r)}r−Cn(u)],u∈[0,1]d, (3)

where . Alternatively, since characterization (1) can equivalently be rewritten as

 {C(u)}r=C(ur),∀u∈[0,1]d,∀r>0,

one could also consider test processes of the form

 Er,n(u)=√n[Cn(ur)−{Cn(u)}r],u∈[0,1]d.

For a given value of , such processes can be used to test the hypothesis

 H0,r:C(u)={C(u1/r)}r∀u∈[0,1]d.

Since , testing for a fixed value of is clearly not equivalent to testing . It follows that tests based on or , with fixed, will only be consistent for copula alternatives for which there exists such that .

In our Monte Carlo experiments, values of smaller than one did not lead to well-behaved tests. Besides, the processes always led to consistently more powerful tests than the processes . For the sake of brevity, we therefore only present the derivation of the tests based on with .

The following result, whose short proof is given in Appendix A, gives the asymptotic behavior of the test process (3) under .

###### Proposition 1.

Suppose that the partial derivatives of satisfy Condition (), and let . Then, under ,

 Dr,n(u)⇝Dr(u)=r{C(u1/r)}r−1C(u1/r)−C(u) (4)

in .

Before suggesting two candidate test statistics based on , let us first explain how, for large , approximate independent copies of can be obtained by means of a multiplier technique initially proposed in Sca05 and RemSca09, and recently revisited in Seg11.

As can be seen from (4), to obtain approximate independent copies of , it is necessary to obtain approximate independent copies of . To estimate the unknown partial derivative , , appearing in the expression of given in (2), we use the estimator defined by

 C[j]n(u)=1u+j,n−u−j,n{Cn(u1,…,uj−1,u+j,n,uj+1,…,ud)−Cn(u1,…,uj−1,u−j,n,uj+1,…,ud)},u∈[0,1]d, (5)

where , and .

This estimator differs slightly from the one initially proposed in RemSca09. It has the advantage of converging in probability to uniformly over if happens to be continuous on instead of only satisfying Condition (). This point is discussed in more detail in Appendix C.

Let us now introduce additional notation. Let be a large integer and let , , , be i.i.d. random variables with mean 0 and variance 1 satisfying , and independent of the data . For any and any , let

 α(k)n(u)=1√nn∑i=1Z(k)i{1(^Ui≤u)−Cn(u)}=1√nn∑i=1(Z(k)i−¯Z(k))1(^Ui≤u),

where . Furthermore, for any and any , let

 C(k)n(u)=α(k)n(u)−d∑j=1C[j]n(u)α(k)n(1,…,1,uj,1,…,1),

and let

 D(k)r,n(u)=r{Cn(u1/r)}r−1C(k)n(u1/r)−C(k)n(u).

The following result, whose proof is given in Appendix A, is at the root of the proposed class of tests.

###### Proposition 2.

Suppose that the partial derivatives of satisfy Condition (), and let . Then, under ,

 (Dr,n,D(1)r,n,…,D(N)r,n)⇝(Dr,D(1)r,…,D(N)r)

in , where are independent copies of the process defined in (4).

As candidate test statistics, we consider Cramér–von Mises functionals of the form

 Sr,n=∫[0,1]d{Dr,n(u)}2du,andTr,n=∫[0,1]d{Dr,n(u)}2dCn(u).

Also, for any , let

 S(k)r,n=∫[0,1]d{D(k)r,n(u)}2du,andT(k)r,n=∫[0,1]d{D(k)r,n(u)}2dCn(u).

The following key result is proved in Appendix B.

###### Proposition 3.

Suppose that the partial derivatives of satisfy Condition (), and let . Then, under ,

 (Sr,n,S(1)r,n,…,S(N)r,n)⇝(Sr,S(1)r,…,S(N)r)

and

 (Tr,n,T(1)r,n,…,T(N)r,n)⇝(Tr,T(1)r,…,T(N)r)

in , where

 Sr=∫[0,1]d{Dr(u)}2duandTr=∫[0,1]d{Dr(u)}2dC(u)

are the weak limits of and , respectively, and and are independent copies of and , respectively.

The previous results suggest computing approximate -values for and as

 1NN∑k=11(S(k)r,n≥Sr,n)and1NN∑k=11(T(k)r,n≥Tr,n),

respectively.

Notice that, when is not true, the processes , , cannot be regarded anymore as approximate independent copies of under because is not anymore a random sample from a c.d.f. , where satisfies for all . This however does not affect the consistency of the procedure with respect to the hypothesis . Indeed, the process can be decomposed as

 Dr,n(u)=√n[{Cn(u1/r)}r−{C(u1/r)}r]−√n{Cn(u)−C(u)}+√n[{C(u1/r)}r−C(u)],u∈[0,1]d.

Whether is false or not, provided Condition () is satisfied, the first and second term will jointly converge weakly to the limit established in the proof of Proposition 1 (see (7) in Appendix A), while, if is false,

On the other hand, from the proof of Proposition 2, it is easy to verify that, provided Condition () is satisfied, and whether is false or not,

 (D(1)r,n,…,D(N)r,n)⇝(D(1)r,…,D(N)r)

in , where are independent copies of the process defined in (4). It follows that the statistics and , as any sensible statistic derived from the process , will be consistent with respect to the hypothesis .

To improve the sensitivity of the proposed tests, given reals , we also consider tests based on statistics of the form

 Sr1,…,rp,n=p∑i=1Sri,nandTr1,…,rp,n=p∑i=1Tri,n. (6)

## 4 Implementation of the tests based on Sr,n and Tr,n

We first discuss the implementation of the test based on . The implementation of the test based on follows immediately after a simple modification.

Given a large integer , we proceed by numerical approximation based on a grid of uniformly spaced points on denoted . Then,

 Sr,n≈1mm∑j=1{Dr,n(wj)}2=nmm∑j=1[{Cn(w1/rj)}r−Cn(wj)]2,

and, for any ,

 S(k)r,n≈1mm∑j=1{D(k)r,n(wj)}2=1mm∑j=1[r{Cn(w1/rj)}r−1C(k)n(w1/rj)−C(k)n(wj)]2.

To efficiently implement the test, first notice that, for any , can be conveniently written as

 C(k)n(u)=1√nn∑i=1(Z(k)i−¯Z(k)){1(^Ui≤u)−d∑l=1C[l]n(u)1(^Uil≤ul)},u∈[0,1]d.

It follows that the can be expressed as

 D(k)r,n(wj)=r{Cn(w1/rj)}r−1C(k)n(w1/rj)−C(k)n(wj)=1√nn∑i=1(Z(k)i−¯Z(k))Mn(i,j)

in terms of a matrix whose -element is

 Mn(i,j)=r{Cn(w1/rj)}r−1{1(^Ui≤w1/rj)−d∑l=1C[l]n(w1/rj)1(^Uil≤w1/rjl)}−{1(^Ui≤wj)−d∑l=1C[l]n(wj)1(^Uil≤wjl)}.

In order to carry out the test based on , it is first necessary to compute the matrix . Then, to compute , it suffices to generate i.i.d. random variates with expectation 0, variance 1, satisfying , and perform simple arithmetic operations involving the centered and the columns of matrix . In the Monte Carlo simulations to be presented in the next section, the are taken from the standard normal distribution.

For the test based on , clearly,

 Tr,n=1nn∑j=1{Dr,n(^Uj)}2andT(k)r,n=1nn∑j=1{D(k)r,n(^Uj)}2,k∈{1,…,N}.

Expressions for implementing the test then immediately follow from those given for and : simply replace by , and by .

## 5 Finite-sample performance

Extensive Monte Carlo experiments were conducted to investigate the level and power of the tests based on and for samples of size , 200, 400 and 800. The values were considered for . We also investigated the finite-sample performance of the tests based on the statistics and defined in (6). Approximate -values for the latter tests can be obtained by proceeding as in the previous section. Several configurations were studied among which and . Data sets of dimension two to five were generated, both from extreme-value and non extreme-value copulas. Given that the most frequently used bivariate exchangeable extreme-value copulas such as the Gumbel–Hougaard, Galambos, Hüsler–Reiss and Student extreme-value copulas show striking similarities for a given degree of dependence (see GenKojNesYan11, for a detailed discussion of this matter), only the Gumbel–Hougaard (GH) and its asymmetric version (aGH) defined using Khoudraji’s device (Kho95; GenGhoRiv98; Lie08) were used in the simulations. Given an exchangeable copula , Khoudraji’s device defines an asymmetric version of it as

 Cθ,λ(u)=u1−λ11…u1−λddCθ(uλ11,…,uλdd),

for all and an arbitrary choice of such that for some . If is an extreme-value copula, then the same is true of . Note that the asymmetric Gumbel–Hougaard (aGH) obtained from Khoudraji’s device is nothing else but the asymmetric logistic model introduced in Taw88; Taw90. In the experiments, the parameter of the asymmetric Gumbel–Hougaard was set to 4. In dimension two, the shape parameter vector was taken equal to , with , so that data generated from the corresponding copulas display various degrees of asymmetry. The corresponding values for Kendall’s tau are about 0.19, 0.34, 0.48 and 0.60, respectively. In dimension three, four and five, was set to , , and , respectively.

As far as non extreme-value copulas are concerned, the Clayton (C), Frank (F), normal (N), with four degrees of freedom (t), and Plackett (P) (for dimension two only) copulas were used in the experiments.

For each of the one-parameter exchangeable families considered in the study (GH, C, F, N, t, P), three values of the parameter were considered. These were chosen so that the bivariate margins of the copulas have a Kendall’s tau of 0.25, 0.50, and 0.75, respectively.

All the tests were carried out at the 5% significance level and empirical rejection rates were computed from 1000 random samples per scenario. For the tests based on , the parameter defined in Section 4 was set to in dimension two, in dimension three, in dimension four, and in dimension five. Smaller and greater values of were also considered but this did not seem to affect the results much.

In most scenarios involving extreme-value copulas, the tests turned out to be globally too conservative, although the agreement with the 5% level seemed to improve as was increased. To attempt to improve the empirical levels of the tests for , we considered several asymptotically negligible ways of rescaling the empirical copula in the expression of the test process (3), while keeping the expressions of the processes , , unchanged. Reasonably good empirical levels were obtained by replacing in the expression of by . With this asymptotically negligible modification, the best results were obtained for and for the tests based on , which consistently outperformed the tests based on . In dimension two, the rejection rates of the tests based on , , , and are reported in Tables 1 and 2.

As can be seen from Table 1, the empirical levels of the selected tests are, overall, reasonably close to the 5% nominal level for and , which, as discussed earlier, corresponds to weak to moderate dependence. The tests remain however too conservative when , although the empirical levels seems globally to improve as increases. An inspection of Table 2 shows that, in terms of power, the tests based on and appear globally more powerful than that based on , although the latter sometimes outperforms the former in the case of weakly dependent data sets. As far as the test based on is concerned, its rejection rates are almost always greater than those of , and sometimes greater than those of .

The previous tests can be compared with the test of extreme-value dependence proposed by GhoKhoRiv98 and improved by BenGenNes09. The rejection rates of the best version of that test, based on a variance estimator denoted , were computed using routines available in the copula R package, and are reported in Table 2. The test based on is more powerful than its competitors when data are generated from an elliptical copula, the gain in power being particularly large for the copula. The proposed tests perform better when data are generated from a Frank or a Plackett copula. For and the Frank copula, the rejection rates of test based on are approximately twice as great as those of the test based on . From the lower right block of Table 2, we also see that, for all tests, the optimal rejection rate is almost reached in all scenarios not involving extreme-value copulas when .

The rejection rates of the test based on for data sets of dimension three, four and five are given in Table 3. As can be seen from the first two horizontal blocks of the table, in the case of weak to moderate dependence, the test appears slightly conservative, overall, although the agreement with the 5% level seems to improve as increases. As in dimension two, the test is the most conservative in the case of strongly dependent data and this phenomenon increases with the dimension. Notice however that, in almost all scenarios under the alternative hypothesis, the power of the test increases as increases. This might be due to the fact that every bivariate margin of a -variate extreme-value copula must be max-stable. Hence, deviations from multivariate max-stability might be easier to detect as the dimension increases. Note finally that, as reaches 800, the optimal rejection rate is almost attained in all scenarios not involving extreme-value copulas.

## 6 Discussion and illustrations

The results of the extensive Monte Carlo experiments partially reported in the previous section suggest that the test based on the statistic can be safely used in dimension two or greater to assess whether data arise from an extreme-value copula. The choice of the statistic is not claimed to be optimal as other candidate test statistics could be considered. In dimension two, the test appears more powerful than the test of BenGenNes09 based on in approximately half of the scenarios under the alternative hypothesis, and is outperformed in the remaining scenarios. In dimension strictly greater than two, the proposed approach is presently, to the best of our knowledge, the only available procedure for testing extreme-value dependence.

As an illustration, we first applied the test based on to the bivariate indemnity payment and allocated loss adjustment expense data studied in FreVal98. These consist of 1466 general liability claims randomly chosen from late settlement lags (among the initial 1500 claims, 34 claims for which the policy limit was reached were ignored). Many studies, including that of BenGenNes09, have concluded that an extreme-value copula is likely to provide an adequate model of the dependence.

Note that these data contain a non negligible number of ties. As is the case for other procedures based on the empirical copula, the presence of ties might significantly affect the tests under study since these were derived under the assumption of continuous margins. To deal somehow satisfactorily with ties, KojYan10 suggested to assign ranks at random in the case of ties when computing pseudo-observations. This was done using the R function rank with its argument ties.method set to "random". The test was then carried out on the resulting pseudo-observations. With the hope that the use of randomization will result in many different configurations for the parts of the data affected by ties, the test based on the pseudo-observations computed with ties.method = "random" was performed 100 times with . The minimum, median and maximum of the obtained approximate -values are 40.7%, 45.9%, and 50.4%, respectively. If the pseudo-observations are computed using mid-ranks, the approximate -value, based on multiplier iterations, drops down to 1.7%. As already observed in other situations, using mid-ranks seems to increase the evidence against the null hypothesis.

As a second example, we considered the uranium exploration data of CooJoh86. The data consist of log-concentrations of seven chemical elements in 655 water samples collected near Grand Junction, Colorado: uranium (U), lithium (Li), cobalt (Co), potassium (K), cesium (Cs), scandium (Sc), and titanium (Ti). BenGenNes09 performed an extensive study of the 21 pairs of variables and suggested that the triples {U,Co,Li}, {U,Li,Ti} and {Ti,Li,Cs} should be investigated for trivariate extreme-value dependence once a multivariate test becomes available. Note that the number of ties in these data is greater than in the insurance data of FreVal98. In particular, the variable Li takes only 90 different values out of 655. For that reason, as previously, we broke the ties at random and repeated the calculations 100 times with . Approximate -values for the test based on are summarized in Table 4. The first three columns give the minimum, median and maximum of the obtained -values. The last column gives the -values computed from the mid-ranks using . As for the insurance data, we see that the use of mid-ranks increases the evidence against the null hypothesis. Based on the randomization approach, we conclude that there is strong evidence against trivariate extreme-value dependence in the triples {U,Co,Li} and {U,Li,Ti}, while there is only marginal evidence against trivariate extreme-value dependence in the triple {Ti,Li,Cs}.

### Acknowledgments

The authors are very grateful to the associate editor and the referees for their constructive and insightful suggestions which helped to clean up a number of errors. The authors also thank Johanna Nešlehová for providing R routines implementing the test based on , and Mark Holmes for very fruitful discussions, as always.

## Appendix A Proofs of Propositions 1 and 2

###### Proof of Proposition 1.

From the limiting behavior of the empirical copula process given in Section 2 and the functional version of Slutsky’s theorem (see e.g., vanWel96, Chap. 3.9), we have that

 (√n[{Cn(u1/r)}r−{C(u1/r)}r]√n{Cn(u)−C(u)})⇝(r{C(u1/r)}r−1C(u1/r)C(u)) (7)

in . The desired result then follows from the continuous mapping theorem (see e.g. vanWel96, Theorem 1.3.6). ∎

###### Proof of Proposition 2.

Let , and notice that

 C[j]n(u)=1n(u+j,n−u−j,n)n∑i=1⎧⎪ ⎪⎨⎪ ⎪⎩1(u−j,n<^Uij≤u+j,n)d∏k=1k≠j1(^Uik≤uk)⎫⎪ ⎪⎬⎪ ⎪⎭,u∈[0,1]d.

Also, for all and all . Hence,

 C[j]n(u)≤1√nn∑i=11{(n+1)u−j,n

where is the rank of among . It follows that

 supu∈[0,1]C[j]n(u)≤supu∈[0,1](n+1)u+j,n