Signed-rank tests for location in the symmetric independent component model

Abstract

The so-called independent component (IC) model states that the observed p -vector X is generated via X = Λ Z + μ , where μ is a p -vector, Λ is a full-rank matrix, and the centered random vector Z has independent marginals. We consider the problem of testing the null hypothesis H 0 : μ = 0 on the basis of i.i.d. observations X 1 , … , X n generated by the symmetric version of the IC model above (for which all ICs have a symmetric distribution about the origin). In the spirit of [M. Hallin, D. Paindaveine, Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks, Annals of Statistics, 30 (2002), 1103–1133], we develop nonparametric (signed-rank) tests, which are valid without any moment assumption and are, for adequately chosen scores, locally and asymptotically optimal (in the Le Cam sense) at given densities. Our tests are measurable with respect to the marginal signed ranks computed in the collection of null residuals Λ ˆ − 1 X i , where Λ ˆ is a suitable estimate of Λ . Provided that Λ ˆ is affine-equivariant, the proposed tests, unlike the standard marginal signed-rank tests developed in [M.L. Puri, P.K. Sen, Nonparametric Methods in Multivariate Analysis, Wiley & Sons, New York, 1971] or any of their obvious generalizations, are affine-invariant. Local powers and asymptotic relative efficiencies (AREs) with respect to Hotelling’s T 2 test are derived. Quite remarkably, when Gaussian scores are used, these AREs are always greater than or equal to one, with equality in the multinormal model only. Finite-sample efficiencies and robustness properties are investigated through a Monte Carlo study.