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

• 成果类型: Article
• 署名作者: Hallin, M; Paindaveine, D
• 署名单位: Universite Libre de Bruxelles; Universite Libre de Bruxelles
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2002
• 页码: 1103-1133
• 关键词: affine-invariant; sign tests; EFFICIENCY

摘要

We propose a family of tests. based on Randles' (1989) concept of interdirections 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, Randles' (1989) multivariate sign test, Peters and Randles' (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 Chernoff-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.