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Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. (English) Zbl 1101.62348

Summary: We propose a family of tests, based on R. H. Randles’ [J. Am. Stat. Assoc. 84, No. 408, 1045–1050 (1989; Zbl 0702.62039)] concept of interdirections and the ranks of pseudo-Mahalanobis distances computed with respect to a multivariate \(M\)-estimator of scatter due to D. E. Tyler [Ann. Stat. 15, 234–251 (1987; Zbl 0628.62053)], 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.

MSC:

62H15 Hypothesis testing in multivariate analysis
62G10 Nonparametric hypothesis testing
62F03 Parametric hypothesis testing
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References:

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