Testing circular symmetry

The author addresses the problem of testing circular data for reflective symmetry about an unknown central direction and proposes a simple omnibus test based on the sample second sine moment about an estimation of this direction. Under quite general conditions, for an underlying distribution which is reflectively symmetric, the large-sample asymptotic distribution of the test statistic is standard normal. Randomization and bootstrap variants of the test are also introduced, and the operating characteristics of different versions of the test are investigated in a Monte Carlo study. The large-sample and bootstrap versions of the test are applied in the analysis of two illustrative examples drawn from the circular statistics literature. L'auteur s'inte'resse a la facon de tester si une loi circulaire est syme'trique par rapport a une direction centrale inconnue; il propose un test omnibus simple fonde sur la valeur observed du deuxieme moment sinusoidal centre par rapport a une estimation de cette direction. Sous des conditions tres generales, il montre que la statistique du test obeit asymptotiquement a une loi normale centree reduite lorsque la distribution des observations est effectivement symeerique. II a recours a des simulations pour etudier le comportement de differentes versions de son test, dont des variantes randomised et bootstrap. Les versions asymptotique et bootstrap du test sont illustrees au moyen de deux jeux de donnees circulaires issus de la litterature sur le sujet.

[1]  Stephen J. Finch,et al.  Robust Univariate Test of Symmetry , 1977 .

[2]  Sándor Csörgő,et al.  Testing for symmetry , 1987 .

[3]  Sumitra Purkayastha,et al.  An almost sure representation of sample circular median , 1995 .

[4]  K. Mardia.  Statistics of Directional Data , 1972 .

[5]  Dana Quade,et al.  U-statistics for skewness or symmetry , 1978 .

[6]  G. S. Watson,et al.  Statistical methods for the analysis of problems in animal orientation and certain biological rhythms , 1966 .

[7]  Richard C Barker,et al.  Using the bootstrap in testing symmetry versus asymmetry , 1987 .

[8]  M. K. Gupta.  AN ASYMPTOTICALLY NONPARAMETRIC TEST OF SYMMETRY , 1967 .

[9]  Dennis D. Boos,et al.  A Test for Asymmetry Associated with the Hodges-Lehmann Estimator , 1982 .

[10]  B. Efron.  Bootstrap Methods: Another Look at the Jackknife , 1979 .

[11]  P. K. Bhattacharya,et al.  Two modified Wilcoxon tests for symmetry about an unknown location parameter , 1982 .

[12]  P. Cabilio,et al.  A simple test of symmetry about an unknown median , 1996 .

[13]  I. D. Hill,et al.  An Efficient and Portable Pseudo‐Random Number Generator , 1982 .

[14]  S. Schach,et al.  Nonparametric symmetry tests for circular distributions , 1969 .

[15]  Douglas G. Simpson,et al.  Robust Direction Estimation , 1992 .

[16]  George E. Policello,et al.  An Asymptotically Distribution-Free Test for Symmetry versus Asymmetry , 1980 .

[17]  Joseph L. Gastwirth,et al.  On the Sign Test for Symmetry , 1971 .

[18]  J. Koziol.  Tests for symmetry about an unknown value based on the empirical distribution function , 1983 .

[19]  Russell V. Lenth,et al.  Robust Measures of Location for Directional Data , 1981 .

[20]  Joseph P. Romano.  Bootstrap and randomization tests of some nonparametric hypotheses , 1989 .

[21]  N. Fisher,et al.  Statistical Analysis of Circular Data , 1993 .

[22]  Joseph P. Romano.  A Bootstrap Revival of Some Nonparametric Distance Tests , 1988 .

[23]  A. I. McLeod,et al.  A Remark on Algorithm as 183. An Efficient and Portable Pseudo‐Random Number Generator , 1985 .

[24]  I. D. Hill,et al.  Correction: Algorithm AS 183: An Efficient and Portable Pseudo-Random Number Generator , 1982 .

[25]  M. A. Stephens.  TECHNIQUES FOR DIRECTIONAL DATA , 1969 .

[26]  Detection of symmetry or lack of 11 and applications , 1988 .

[27]  P. Guttorp,et al.  Robustness of Estimators for Directional Data , 1988 .

[28]  P. E. Jupp,et al.  Sobolev Tests for Symmetry of Directional Data , 1983 .

[29]  A. Pewsey.  The wrapped skew-normal distribution on the circle , 2000 .