Rank Tests for Elliptical Graphical Modeling

Paindaveine, Davy; Verdebout, Thomas

Journal de la société française de statistique, Tome 153 (2012) no. 1, pp. 82-100.

En réaction aux hypothèses gaussiennes restrictives qui accompagnent le plus souvent les modèles graphiques, Vogel et Fried [ 17 ] ont récemment introduit des modèles graphiques elliptiques, qui prévoient que les variables suivent conjointement une distribution elliptique. Le présent travail introduit une classe de tests de rangs dans le contexte de ces modèles graphiques elliptiques. Ces tests sont valides sous une densité elliptique quelconque, et en particulier ne requièrent aucune hypothèse de moment. Ils sont localement et asymptotiquement optimaux sous des densités correctement spécifiées. Leurs propriétés asymptotiques sont étudiées à la fois sous l’hypothèse nulle et sous des suites de contre-hypothèses locales. Leurs efficacités asymptotiques relatives par rapport à leurs compétiteurs pseudo-gaussiens sont calculées, ce qui permet de montrer que, lorsqu’ils sont basés sur des scores gaussiens, les tests de rangs proposés dominent uniformément les tests pseudo-gaussiens au sens de Pitman. Les résultats asymptotiques sont confirmés par une étude de Monte-Carlo.

As a reaction to the restrictive Gaussian assumptions that are usually part of graphical models, Vogel and Fried [ 17 ] recently introduced elliptical graphical models, in which the vector of variables at hand is assumed to have an elliptical distribution. The present work introduces a class of rank tests in the context of elliptical graphical models. The proposed tests are valid under any elliptical density, and in particular do not require any moment assumption. They achieve local and asymptotic optimality under correctly specified densities. Their asymptotic properties are investigated both under the null and under sequences of local alternatives. Asymptotic relative efficiencies with respect to the corresponding pseudo-Gaussian competitors are derived, which allows to show that, when based on normal scores, the proposed rank tests uniformly dominate the pseudo-Gaussian tests in the Pitman sense. The asymptotic results are confirmed through a Monte-Carlo study.

Keywords: Conditional independence, Graphical models, Local asymptotic normality, Pseudo-Gaussian tests, Rank tests, Scatter matrix, Signed ranks
Mot clés : Indépendance conditionnelle, Matrice de scatter, Modèles graphiques, Normalité locale asymptotique, Rangs signés, Tests de rangs, Tests pseudo-gaussiens

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