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Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks

par Hallin, Marc ;Paindaveine, Davy
Référence Annals of Statistics, 30, 4, page (1103-1133)
Publication Publié, 2002

Article révisé par les pairs

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Résumé :

We propose a family of tests, based on Randies' (1989) concept of inter-directions and the ranks of pseudo-Mahalanobis distances computed with respect to a multivariate M-estimator of scatter due to Tyler (1987), for the multivariate one-sample problem under elliptical symmetry. These tests, which generalize the univariate signed-rank tests, are affine-invariant. Depending on the score function considered (van der Waerden, Laplace, . . .), they allow for locally asymptotically maximin tests at selected densities (multivariate normal, multivariate double-exponential, . . .). Local powers and asymptotic relative efficiencies are derived - with respect to Hotelling's test, Randies' (1989) multivariate sign test, Peters and Randies' (1990) Wilcoxon-type test, and with respect to the Oja median tests. We, moreover, extend to the multivariate setting two famous univariate results: the traditional Chemoff-Savage (1958) property, showing that Hotelling's traditional procedure is uniformly dominated, in the Pitman sense, by the van der Waerden version of our tests, and the celebrated Hodges-Lehmann (1956) ".864 result," providing, for any fixed space dimension k, the lower bound for the asymptotic relative efficiency of Wilcoxon-type tests with respect to Hotelling's. These asymptotic results are confirmed by a Monte Carlo investigation, and application to a real data set.