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Le Cam optimal tests for symmetry against Ferreira and Steel's general skewed distributions

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  • Christophe Ley
  • Davy Paindaveine

Abstract

When testing symmetry of a univariate density, (parametric classes of) densities skewed by means of the general probability transform introduced in Ferreira and Steel [A constructive representation of univariate skewed distributions, J. Amer. Statist. Assoc. 101 (2006), pp. 823–829] are appealing alternatives. This paper first proposes parametric tests of symmetry (about a specified centre) that are locally and asymptotically optimal (in the Le Cam sense) against such alternatives. To improve on these parametric tests, which are valid under well-specified density types only, we turn them into semiparametric tests, either by using a standard studentisation approach or by resorting to the invariance principle. The second approach leads to robust yet efficient signed-rank tests, which include the celebrated sign and Wilcoxon tests as special cases, and turn out to be Le Cam optimal irrespective of the underlying original symmetric density. Optimality, however, is only achieved under well-specified ‘skewing mechanisms’, and we therefore evaluate the overall performances of our tests by deriving their asymptotic relative efficiencies with respect to the classical test of skewness. A Monte-Carlo study confirms the asymptotic results.

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  • Christophe Ley & Davy Paindaveine, 2009. "Le Cam optimal tests for symmetry against Ferreira and Steel's general skewed distributions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(8), pages 943-967.
    • Handle: RePEc:taf:gnstxx:v:21:y:2009:i:8:p:943-967
    DOI: 10.1080/10485250902971765
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    Cited by:

    1. Christophe Ley, 2014. "Flexible Modelling in Statistics: Past, present and Future," Working Papers ECARES ECARES 2014-42, ULB -- Universite Libre de Bruxelles.
    2. Klein, Ingo & Doll, Monika, 2018. "Tests on asymmetry for ordered categorical variables," FAU Discussion Papers in Economics 03/2018, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics.
    3. Bacci, Silvia & Bartolucci, Francesco, 2014. "Mixtures of equispaced normal distributions and their use for testing symmetry with univariate data," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 262-272.
    4. Bojana Milošević & Marko Obradović, 2016. "Two-dimensional Kolmogorov-type goodness-of-fit tests based on characterisations and their asymptotic efficiencies," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(2), pages 413-427, June.
    5. Ley, Christophe & Paindaveine, Davy, 2010. "Multivariate skewing mechanisms: A unified perspective based on the transformation approach," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1685-1694, December.
    6. Christopher Partlett & Prakash Patil, 2017. "Measuring asymmetry and testing symmetry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 429-460, April.

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