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Depth-based runs tests for bivariate central symmetry

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Abstract
McWilliams (J Am Stat Assoc 85:1130-1133, 1990) introduced a nonparametric procedure based on runs for the problem of testing univariate symmetry about the origin (equivalently, about an arbitrary specified center). His procedure first reorders the observations according to their absolute values, then rejects the null when the number of runs in the resulting series of signs is too small. This test is universally consistent and enjoys good robustness properties, but is unfortunately limited to the univariate setup. In this paper, we extend McWilliams' procedure into tests of bivariate central symmetry. The proposed tests first reorder the observations according to their statistical depth in a symmetrized version of the sample, then reject the null when an original concept of simplicial runs is too small. Our tests are affine invariant and have good robustness properties. In particular, they do not require any finite moment assumption. We derive their limiting null distribution, which establishes their asymptotic distribution freeness. We study their finite-sample properties through Monte Carlo experiments and conclude with some final comments.
Keywords
MULTIVARIATE RANK-TESTS, Asymmetric distributions, Central symmetry testing, Multivariate runs, Statistical depth

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Citation

Please use this url to cite or link to this publication:

Chicago
Dyckerhoff, Rainer, Christophe Ley, and Davy Paindaveine. 2015. “Depth-based Runs Tests for Bivariate Central Symmetry.” Annals of the Institute of Statistical Mathematics 67 (5): 917–941.
APA
Dyckerhoff, R., Ley, C., & Paindaveine, D. (2015). Depth-based runs tests for bivariate central symmetry. ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS, 67(5), 917–941.
Vancouver
1.
Dyckerhoff R, Ley C, Paindaveine D. Depth-based runs tests for bivariate central symmetry. ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS. 2015;67(5):917–41.
MLA
Dyckerhoff, Rainer, Christophe Ley, and Davy Paindaveine. “Depth-based Runs Tests for Bivariate Central Symmetry.” ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS 67.5 (2015): 917–941. Print.
@article{8515415,
  abstract     = {McWilliams (J Am Stat Assoc 85:1130-1133, 1990) introduced a nonparametric procedure based on runs for the problem of testing univariate symmetry about the origin (equivalently, about an arbitrary specified center). His procedure first reorders the observations according to their absolute values, then rejects the null when the number of runs in the resulting series of signs is too small. This test is universally consistent and enjoys good robustness properties, but is unfortunately limited to the univariate setup. In this paper, we extend McWilliams' procedure into tests of bivariate central symmetry. The proposed tests first reorder the observations according to their statistical depth in a symmetrized version of the sample, then reject the null when an original concept of simplicial runs is too small. Our tests are affine invariant and have good robustness properties. In particular, they do not require any finite moment assumption. We derive their limiting null distribution, which establishes their asymptotic distribution freeness. We study their finite-sample properties through Monte Carlo experiments and conclude with some final comments.},
  author       = {Dyckerhoff, Rainer and Ley, Christophe and Paindaveine, Davy},
  issn         = {0020-3157},
  journal      = {ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS},
  keyword      = {MULTIVARIATE RANK-TESTS,Asymmetric distributions,Central symmetry testing,Multivariate runs,Statistical depth},
  language     = {eng},
  number       = {5},
  pages        = {917--941},
  title        = {Depth-based runs tests for bivariate central symmetry},
  url          = {http://dx.doi.org/10.1007/s10463-014-0480-y},
  volume       = {67},
  year         = {2015},
}

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