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

Quessy, Jean-François; Bellerive, Rachelle

Journal de la société française de statistique, Numéro spécial sur les copules, Tome 154 (2013) no. 1, pp. 78-101.

Les lois méta-elliptiques sont des modèles statistiques multivariés dans lesquels la structure de dépendance est gouvernée par une copule elliptique et où les distributions marginales sont arbitraires. Dans cet article, des tests d’adéquation sont proposés afin de construire un modèle méta-elliptique approprié pour des données multidimensionnelles. Alors que le choix des marges peut se faire via des tests d’adéquation classiques, comment sélectionner une copule elliptique adéquate est moins clair. Pour combler ce manque, des méthodes d’adéquation formelles sont développées ici autour de la partie radiale qui caractérise une loi elliptique. L’idée centrale consiste à estimer sa fonction de répartition univariée à partir d’un pseudo-échantillon qui découle des observations multivariées originales. Ensuite, on utilise comme statistique de test la distance de Cramér–von Mises entre cet estimateur non-paramétrique et sa version attendue sous l’hypothèse nulle. Une $p$-valeur approximative est obtenue d’une application du bootstrap paramétrique. La méthode est généralisée au cas où le générateur elliptique est à paramètres inconnus en utilisant un critère à distance minimale. Bien que le comportement asymptotique des tests ne soit pas étudié ici, des simulations Monte–Carlo indiquent que les méthodes possèdent de belles propriétés échantillonnales en termes de seuil et de puissance. Les techniques sont illustrées sur les jeux de données “Danish fire insurance”, “Upper Mississippi river”, “Oil currency” et “Uranium exploration”.

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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