On nonparametric tests of multivariate meta-ellipticity
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On nonparametric tests of multivariate meta-ellipticity Jean-François Quessy1 Received: 6 November 2019 / Revised: 30 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract A statistical procedure to determine if the dependence structure of a multivariate random vector belongs or not to the general class of elliptical copulas has been proposed by Jaser et al. (Depend Model 5:330–353, 2017). Their test exploits the fact that when the copula of a multivariate population is elliptical, the theoretical Kendall and Blomqvist dependence measures of each pair are the same. Under a setup where the marginal distributions are known, they based their decision rule on the asymptotic distribution of the proposed test statistic, which is chi-squared. In this paper, the restrictive assumption of known marginals is relaxed by the use of ranks. In addition, new test statistics are proposed and their p-values are computed from suitably adapted bootstrap replicates based on the form of their limit under the null hypothesis. Unlike Jaser et al.’s test, the proposed procedures keep their nominal level well when the dimension exceeds two. It is also shown that the new tests have good power properties against several types of alternatives to copula ellipticity. Keywords Blomqvist’s beta · Kendall’s tau · Multiplier bootstrap · Shape hypotheses
1 Introduction Elliptical distributions are very useful multivariate models, mainly because they allow for various kinds of tail behavior and heterogeneous levels of dependence for the pairs. According for instance to Cambanis et al. (1981) and Fang et al. (1990), a random vector X = (X 1 , . . . , X d ) follows an elliptical distribution if it admits the stochastic representation
X = μ + G A U,
B 1
(1)
Jean-François Quessy [email protected] Département de mathématiques et d’informatique, Université du Québec à Trois-Rivières, P.B. 500, Trois-Rivières G9A 5H7, Canada
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J.-F. Quessy
where μ ∈ Rd is the mean vector, A ∈ Rd×m is such that Σ = A A ∈ Rd×d is symmetric with rank(Σ) = m, U is uniformly distributed on the unit sphere of Rd , and G is a non-negative random variable, called the radial part, that is independent of U. Special cases of this general representation include the d-dimensional normal, Student, Laplace and Pearson type II distributions, to name only a few. By construction, the univariate marginals F1 , . . . , Fd of a given elliptical distribution belong to the same location-scale elliptical family. Meta-elliptical distributions are much more flexible since they allow for arbitrary marginals. Formally, Y = (Y1 , . . . , Yd ) is said to have a meta-elliptical distribution if for some increasing functions η1 , . . . , ηd : R → R, (η1 (Y1 ), . . . , ηd (Yd )) admits a stochastic representation of the form (1). Invoking the well-known Sklar representation, one can then write the distribution of Y as P(Y1 ≤ y1 , . . . , Yd ≤ yd ) = C{F1 ◦ η1 (y1 ), . . . , Fd ◦ ηd (yd )}, where C : [0, 1]d → [0, 1] belongs to the family of elliptical copul
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