Rank Tests for Elliptical Graphical Modeling
[Tests de Rangs pour les Modèles Graphiques Elliptiques]
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 
@article{JSFS_2012__153_1_82_0,
     author = {Paindaveine, Davy and Verdebout, ~Thomas},
     title = {Rank {Tests} for {Elliptical} {Graphical} {Modeling}},
     journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique},
     pages = {82--100},
     publisher = {Soci\'et\'e fran\c{c}aise de statistique},
     volume = {153},
     number = {1},
     year = {2012},
     mrnumber = {2930292},
     zbl = {1316.62076},
     language = {en},
     url = {http://archive.numdam.org/item/JSFS_2012__153_1_82_0/}
}
TY  - JOUR
AU  - Paindaveine, Davy
AU  - Verdebout,  Thomas
TI  - Rank Tests for Elliptical Graphical Modeling
JO  - Journal de la société française de statistique
PY  - 2012
SP  - 82
EP  - 100
VL  - 153
IS  - 1
PB  - Société française de statistique
UR  - http://archive.numdam.org/item/JSFS_2012__153_1_82_0/
LA  - en
ID  - JSFS_2012__153_1_82_0
ER  - 
%0 Journal Article
%A Paindaveine, Davy
%A Verdebout,  Thomas
%T Rank Tests for Elliptical Graphical Modeling
%J Journal de la société française de statistique
%D 2012
%P 82-100
%V 153
%N 1
%I Société française de statistique
%U http://archive.numdam.org/item/JSFS_2012__153_1_82_0/
%G en
%F JSFS_2012__153_1_82_0
Paindaveine, Davy; Verdebout,  Thomas. Rank Tests for Elliptical Graphical Modeling. Journal de la société française de statistique, Tome 153 (2012) no. 1, pp. 82-100. http://archive.numdam.org/item/JSFS_2012__153_1_82_0/

[1] Anderson, T W An Introduction to Multivariate Statistical Analysis (Wiley Series in Probability and Statistics), Wiley-Interscience, 2003 | MR | Zbl

[2] Bentler, PM; Berkane, M Greatest lower bound to the elliptical theory kurtosis parameter, Biometrika, Volume 73 (1986), pp. 240-241 | DOI | MR | Zbl

[3] Bentler, PM Some contributions to efficient statistics in structural models: Specification and estimation of moment structures, Psychometrika, Volume 48 (1983), pp. 493-517 | MR | Zbl

[4] Hallin, M; Oja, H; Paindaveine, D Semiparametrically efficient rank-based inference for shape II. Optimal R-estimation of shape, The Annals of Statistics, Volume 34 (2006), pp. 2757-2789 | MR | Zbl

[5] Hallin, M; Paindaveine, D Semiparametrically efficient rank-based inference for shape I. optimal rank-based tests for sphericity, The Annals of Statistics, Volume 34 (2006), pp. 2707-2756 | MR | Zbl

[6] Hallin, M; Paindaveine, D Optimal rank-based tests for homogeneity of scatter, Annals of Statistics, Volume 36 (2008), pp. 1261-1298 | MR | Zbl

[7] Hallin, M; Paindaveine, D; Verdebout, T Optimal rank-based testing for principal components, The Annals of Statistics, Volume 38 (2010), pp. 3245-3299 | MR | Zbl

[8] Hettmansperger, TP; Randles, RH A practical affine equivariant multivariate median, Biometrika, Volume 89 (2002), pp. 851-860 | DOI | MR | Zbl

[9] Ilmonen, P; Paindaveine, D Semiparametrically efficient inference based on signed ranks in symmetric independent component models, Annals of Statistics, Volume 39 (2011), pp. 2448-2476 | MR | Zbl

[10] Kreiss, JP On adaptive estimation in stationary ARMA processes, The Annals of Statistics, Volume 15 (1987), pp. 112-133 | MR | Zbl

[11] Le Cam, L Asymptotic Methods in Statistical Decision Theory (Springer Series in Statistics), Springer, New York, 1986 | MR | Zbl

[12] Li, H; Gui, J Gradient directed regularization for sparse Gaussian concentration graphs, with applications to inference of genetic networks, Biostatistics, Volume 7 (2005), pp. 302-317 | Zbl

[13] Magwene, PM; Kim, J Estimating genomic coexpression networks using first-order conditional independence, Genome Biology, Volume 5 (2004) | DOI

[14] Paindaveine, D A Chernoff-Savage result for shape: on the non-admissibility of pseudo-Gaussian methods, Journal of Multivariate Analysis, Volume 97 (2006), pp. 2206-2220 | MR | Zbl

[15] Rao, C R; Mitra, Sujit K Generalized Inverse of Matrices and Its Applications, John Wiley & Sons Inc, New York, 1971 | MR | Zbl

[16] Toh, H; Horimoto, K Inference of a genetic network by a combined approach of cluster analysis and graphical Gaussian modeling, Bioinformatics, Volume 18 (2002), pp. 287-297

[17] Vogel, D; Fried, R Elliptical graphical modelling, Biometrika, Volume 98 (2011), pp. 935-951 | MR | Zbl