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

Jean-François Quessy, Rachelle Bellerive


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.

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Ce travail est autorisé sous licence avec la Licence de paternité Creative Commons 3.0.

SFdS / SMF - Journal de la Société Française de Statistique - ISSN 2102-6238