Checking the adequacy of the multivariate semiparametric location shift model

  • Authors:

  • N. Henze;B. Klar;L. X. Zhu

  • Affiliations:

  • Institute of Mathematical Stochastics, University of Karlsruhe, Germany;Institute of Mathematical Stochastics, University of Karlsruhe, Germany;Department of Statistics and Actuarial Sciences, University of Hong Kong, Hong Kong, China and East China Normal University, China

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2005

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Abstract

Let X, X1, ..., Xm, ..., Y, Y1, ..., Yn, ... be independent d-dimensional random vectors, where the Xj are i.i.d, copies of X, and the Yk are i.i.d, copies of Y. We study a class of consistent tests for the hypothesis that Y has the same distribution as X + µ for some unspecified µ ∈ Rd. The test statistic L is a weighted integral of the squared modulus of the difference of the empirical characteristic functions of X1 + µ, ..., Xm + µ and Y1, ..., Yn, where µ is an estimator of µ. An alternative representation of L is given in terms of an L2-distance between two nonparametric density estimators. The finite-sample and asymptotic null distribution of L is independent of µ. Carried out as a bootstrap or permutation procedure, the test is asymptotically of a given size, irrespective of the unknown underlying distribution. A large-scale simulation study shows that the permutation procedure performs better than the bootstrap.