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

Abstract

a b s t r a c tThe so-called independent component (IC) model states that the observed p-vector X is generated via X = ΛZ + 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 residualŝ Λ −1 X i , whereΛ is a suitable estimate of Λ. Provided thatΛ 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.

Invariant Co-Ordinate Selection

“…Beyond the role that it plays in detecting departures from ellipticity, invariant co‐ordinate selection (ICS) is potentially useful to choose a proper model for the data at hand among the many multivariate models that are available in the literature: (mixtures of) elliptical models, the independent component (IC) models of (see also Nordhausen et al . (2009b)), skew elliptical models (see, for example, Genton (2004)), etc.…”

The Growing Hierarchical Neural Gas Self-Organizing Neural Network

“…Hence, a graph is not affected by homogeneous scalings, translations, reflections, and rotations. This is a very desirable feature of any multidimensional data analysis method [29]- [31], and in particular of self-organizing neural networks [17]. This property is also fulfilled by the overall GHNG trees.…”

Comparison of Multivariate Means across Groups with Ordinal Dependent Variables: A Monte Carlo Simulation Study

“…As mentioned above, this can be done using either inner or outer standardization. In the context of MANOVA, the use of outer standardization is appropriate when the dependent variables are on the same scale, whereas when they are not on the same scale, inner standardization should be used [45]. Thus, the decision as to which standardization approach the researcher will use should be based on the nature of the data itself.…”