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• W1995524554 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 p-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 Ghoudi et al. (1998) recently revisited by Ben Ghorbal et al. (2009). The proposed procedures are illustrated on bivariate financial data and trivariate geological data. The Canadian Journal of Statistics 39: 703–720; 2011. © 2011 Statistical Society of Canada Les auteurs proposent des tests, pour grands échantillons, d'appartenance à la classe des copules de valeurs extrêmes multivariées en partant du fait que ces structures de dépendance sont caractérisées par la propriété de max-stabilité. Les deux ingrédients clés de l'approche proposée sont la copule empirique d'une part, et une technique à base de multiplicateurs permettant le calcul de valeurs approchées d'autre part. Les auteurs démontrent la validité asymptotique de la méthode proposée et étudient empiriquement la puissance d'un grand nombre de versions du test à l'aide de simulations de Monte Carlo en dimension deux, trois, quatre et cinq. Dans le cas bivarié, les auteurs comparent les meilleures versions du test avec le test de Ghoudi et al.(1998) récemment amélioré par Ben Ghorbal et al. (2009). Ils illustrent leur propos avec des données financières bivariées et des données géologiques trivariées. La revue canadienne de statistique 39: 703–720; 2011. © 2011 Société statistique du Canada Given a random sample from c.d.f. , it is of interest in many applications to test whether the unknown copula C belongs to the class of extreme-value copulas. A first solution to this problem was proposed in the bivariate case by Ghoudi, Khoudraji, & Rivest (1998) who derived a test based on the bivariate probability integral transformation. The suggested approach was recently improved by Ben Ghorbal, Genest, & Nešlehová (2009) 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 p-values for candidate test statistics. This is the second key element of the proposed approach and is based on the seminal work of Scaillet (2005) and Rémillard & Scaillet (2009), revisited recently in Segers (2011). 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 Segers (2011). 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 Frees & Valdez (1998), and trivariate extreme-value dependence in the uranium exploration data of Cook & Johnson (1986). The following notational conventions are adopted in the sequel. The arrow ‘↝’ denotes weak convergence in the sense of Definition 1.3.3 in van der Vaart & Wellner (2000), and represents the space of all bounded real-valued functions on [0,1]d equipped with the uniform metric. Also, for any and any r > 0, we adopt the notation . Furthermore, the set of extreme-value copulas, that is, copulas satisfying (1), is denoted by . Note finally that all the tests studied in this work are implemented in the R package copula (Kojadinovic & Yan, 2010) available on the Comprehensive R Archive Network (R Development Core Team, 2011). () for any , C[j] exists and is continuous on the set . Since , testing H0,r for a fixed value of r is clearly not equivalent to testing H0. It follows that tests based on or , with r fixed, will only be consistent for copula alternatives for which there exists such that . In our Monte Carlo experiments, values of r 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 r ≥ 1. The following result, whose short proof is given in Appendix A, gives the asymptotic behavior of the test process (3) under H0,r. Before suggesting two candidate test statistics based on , let us first explain how, for large n, approximate independent copies of can be obtained by means of a multiplier technique initially proposed in Scaillet (2005) and Rémillard & Scaillet (2009), and recently revisited in Segers (2011). This estimator differs slightly from the one initially proposed in Rémillard & Scaillet (2009). It has the advantage of converging in probability to C[j] uniformly over [0,1]d if C[j] happens to be continuous on [0,1]d instead of only satisfying Condition (). This point is discussed in more detail in Appendix C. The following result, whose proof is given in Appendix A, is at the root of the proposed class of tests. The following key result is proved in Appendix B. We first discuss the implementation of the test based on . The implementation of the test based on follows immediately after a simple modification. In order to carry out the test based on , it is first necessary to compute the n × m matrix Mn. Then, to compute , it suffices to generate n i.i.d. random variates with expectation 0, variance 1, satisfying , and perform simple arithmetic operations involving the centered and the columns of matrix Mn. In the Monte Carlo simulations to be presented in the next section, the are taken from the standard normal distribution. Expressions for implementing the test then immediately follow from those given for and : simply replace m by n, and wj by . As far as non extreme-value copulas are concerned, the Clayton (C), Frank (F), normal (N), t 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 1,000 random samples per scenario. For the tests based on , the parameter m defined in Section 4 was set to 442 in dimension two, 133 in dimension three, 74 in dimension four, and 55 in dimension five. Smaller and greater values of m 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 n 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 Cn 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 T3,n, T4,n, T5,n, and T3,4,5,n 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 τ = 0.75, although the empirical levels seems globally to improve as n increases. An inspection of Table 2 shows that, in terms of power, the tests based on T4,n and T5,n appear globally more powerful than that based on T3,n, although the latter sometimes outperforms the former in the case of weakly dependent data sets. As far as the test based on T3,4,5,n is concerned, its rejection rates are almost always greater than those of T4,n, and sometimes greater than those of T5,n. The previous tests can be compared with the test of extreme-value dependence proposed by Ghoudi, Khoudraji, & Rivest (1998) and improved by Ben Ghorbal et al. (2009). 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 t copula. The proposed tests perform better when data are generated from a Frank or a Plackett copula. For n = 100 and the Frank copula, the rejection rates of test based on T3,4,5,n 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 n = 800. The rejection rates of the test based on T3,4,5,n 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 n 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 d increases. This might be due to the fact that every bivariate margin of a d-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 n reaches 800, the optimal rejection rate is almost attained in all scenarios not involving extreme-value copulas. The results of the extensive Monte Carlo experiments partially reported in the previous section suggest that the test based on the statistic T3,4,5,n can be safely used in dimension two or greater to assess whether data arise from an extreme-value copula. The choice of the statistic T3,4,5,n 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 Ben Ghorbal et al. (2009) 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 T3,4,5,n to the bivariate indemnity payment and allocated loss adjustment expense data studied in Frees & Valdez (1998). These consist of 1,466 general liability claims randomly chosen from late settlement lags (among the initial 1,500 claims, 34 claims for which the policy limit was reached were ignored). Many studies, including that of Ben Ghorbal et al. (2009), 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, Kojadinovic & Yan (2010) 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 N = 1,000. The minimum, median, and maximum of the obtained approximate p-values are 40.7%, 45.9%, and 50.4%, respectively. If the pseudo-observations are computed using mid-ranks, the approximate p-value, based on N = 10,000 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 Cook & Johnson (1986). 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). Ben Ghorbal et al. (2009) 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 Frees & Valdez (1998). 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 N = 1,000. Approximate p-values for the test based on T3,4,5,n are summarized in Table 4. The first three columns give the minimum, median and maximum of the obtained p-values. The last column gives the p-values computed from the mid-ranks using N = 10,000. 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}. 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. Funding for Johan Segers was provided by IAP research network grant P6/03 of the Belgian government (Belgian Science Policy) and by “Projet d'actions de recherche concertées” number 07/12/002 of the Communauté française de Belgique, granted by the Académie universitaire de Louvain. In order to prove the joint weak convergence of , we first show a lemma. Let A be the space of bounded, Borel measurable functions on [0,1]d and let B be the space of c.d.f.s of finite Borel measures on [0,1]d. Define by and denote for . The topologies on A and B are the ones induced by uniform convergence. The topology on A × B, AN+1 or is the product topology. Lemma 1. The map ϕ is continuous at each pair (a0, b0) of A × B such that the functions a0 and b0 are continuous on [0,1]d. We treat the two terms on the right-hand side of the previous inequality separately. Second, as , we have in the topology of weak convergence of finite Borel measures. By continuity of the function a0, this implies , as required. Proof of Proposition 3. The fact that jointly converge weakly to independent copies of the same limit is an immediate consequence of Proposition 2 and the continuous mapping theorem. Let and . Copulas being continuous, C belongs to B0. From Proposition 1, we have that belongs to A0 with probability one since the same is true for defined in (2). The limiting processes also belong to A0 with probability one since they are independent copies of . It is easy to verify that, for fixed and n sufficiently large, and coincide on , and hence, from Lemma 2 below, if C[j] is continuous on the set Vj defined in Condition (), both estimators converge in probability to C[j] uniformly on . It could be argued that the following is a desirable property of an estimator of C[j]: if C[j] happens to be continuous on [0,1]d instead of Vj, the estimator should converge in probability to C[j] uniformly on [0,1]d. This property is satisfied by as is verified in Lemma 2 below. It is however not satisfied by since the latter estimator does not converge pointwise in probability at points u of [0,1]d such that uj = 0 or uj = 1. The term on the right converges to zero as n tends to infinity because C[1] is uniformly continuous on the set ." @default.
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• W1995524554 title "Large-sample tests of extreme-value dependence for multivariate copulas" @default.
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