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Principal component decomposition of non-parametric tests
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  • Published: June 1995

Principal component decomposition of non-parametric tests

  • Arnold Janssen1 

Probability Theory and Related Fields volume 101, pages 193–209 (1995)Cite this article

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Summary

Let ϕ denote an arbitrary non-parametric unbiased test for a Gaussian shift given by an infinite dimensional parameter space. Then it is shown that the curvature of its power function has a principal component decomposition based on a Hilbert-Schmidt operator. Thus every test has reasonable curvature only for a finite number of orthogonal directions of alternatives. As application the two-sided Kolmogorov-Smirnov goodnessof-fit test is treated. We obtain lower bounds for their local asymptotic relative efficiency. They converge to one as α↓0 for the directionh 0(u)=sign(2u−1) of the gradient of the median test. These results are analogous to earlier results of Hájek and Šidák for one-sided Kolmogorov-Smirnov tests.

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Authors and Affiliations

  1. Mathematisches Institut, Heinrich-Heine-Universität Düsseldorf, Universitätsstrasse 1, D-40225, Düsseldorf, Germany

    Arnold Janssen

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  1. Arnold Janssen
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Janssen, A. Principal component decomposition of non-parametric tests. Probab. Th. Rel. Fields 101, 193–209 (1995). https://doi.org/10.1007/BF01375824

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  • Received: 01 December 1992

  • Revised: 21 July 1994

  • Issue Date: June 1995

  • DOI: https://doi.org/10.1007/BF01375824

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Mathematics Subject Classification (1991)

  • 62G10
  • 62G20
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