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

Author

Listed:
  • Nordhausen, Klaus
  • Oja, Hannu
  • Paindaveine, Davy

Abstract

The so-called independent component (IC) model states that the observed p-vectorX is generated via X=[Lambda]Z+[mu], where [mu] is a p-vector, [Lambda] is a full-rank matrix, and the centered random vector Z has independent marginals. We consider the problem of testing the null hypothesis on the basis of i.i.d.observations X1,...,Xn 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 , where is a suitable estimate of[Lambda]. Provided that 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 T2 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.

Suggested Citation

  • Nordhausen, Klaus & Oja, Hannu & Paindaveine, Davy, 2009. "Signed-rank tests for location in the symmetric independent component model," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 821-834, May.
  • Handle: RePEc:eee:jmvana:v:100:y:2009:i:5:p:821-834
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    Citations

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    Cited by:

    1. Hannu Oja & Davy Paindaveine & Sara Taskinen, 2009. "Parametric and nonparametric test for multivariate independence in IC models," Working Papers ECARES 2009_018, ULB -- Universite Libre de Bruxelles.
    2. Ilmonen, Pauliina & Nevalainen, Jaakko & Oja, Hannu, 2010. "Characteristics of multivariate distributions and the invariant coordinate system," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1844-1853, December.
    3. Davy Paindaveine & Germain Van Bever, 2017. "Halfspace Depths for Scatter, Concentration and Shape Matrices," Working Papers ECARES ECARES 2017-19, ULB -- Universite Libre de Bruxelles.
    4. David E. Tyler & Frank Critchley & Lutz Dümbgen & Hannu Oja, 2009. "Invariant co-ordinate selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(3), pages 549-592.

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