Statistical Procedures for the Selection of a Multidimensional Meta-elliptical Distribution

Résumé

Meta-elliptical distributions are multivariate statistical models in which the dependence structure is governed by an elliptical copula and where the marginal distributions are arbitrary. In this paper, goodness-of-fit tests are proposed for the construction of an appropriate meta-elliptical model for multidimensional data. While the choice of the marginal distributions can be guided by classical goodness-of-fit testing, how to select an adequate elliptical copula is less clear. In order to fill this gap, formal copula goodness-of-fit methodologies are developed here around the radial part that characterizes an elliptical distribution. The key idea consists in estimating its univariate distribution function from a pseudo-sample derived from the original multivariate observations. Then, a Cramér–von Mises distance between this non-parametric estimator and the expected parametric version under the null hypothesis is used as a test statistic. An approximate p-value is obtained from an application of the parametric bootstrap. The method is extended to the case where the elliptical generator has unknown parameters using a minimum-distance criterion. While a careful investigation of the asymptotic behavior of the tests is not presented here, Monte–Carlo simulations indicate that the methods have good sample properties in terms of size and power. The techniques are illustrated on the Danish fire insurance, Upper Mississippi river, Oil currency and Uranium exploration data sets.