Testing for symmetry

SUMMARY We introduce the characteristic symmetry function, based on the characteristic function of the underlying distribution, whose behaviour is indicative of symmetry or its absence. A statistic is proposed for testing symmetry about an unspecified centre, derived from the empirical characteristic symmetry function. The statistic is readily computible, it utilizes information in the empirical characteristic function over an interval, and does not require the estimation of the centre of symmetry. Under general symmetry the asymptotic null distribution of the statistic is folded normal. The empirical power for selected alternatives is studied by a small-scale simulation and a numerical illustration is given.

[1]  S. Csörgo.  Multivariate empirical characteristic functions , 1981 .

[2]  J. K. Ord,et al.  The Discrete Student's $t$ Distribution , 1968 .

[3]  Kjell A. Doksum,et al.  Plots and tests for symmetry , 1977 .

[4]  P. V. Rao,et al.  Tests of symmetry based on Cramer-von Mises statistics , 1977 .

[5]  C. Heathcote,et al.  Some results concerning symmetric distributions , 1982, Bulletin of the Australian Mathematical Society.

[6]  X. Fernique,et al.  Régularité de processus gaussiens , 1971 .

[7]  C. R. Heathcote,et al.  Linear regression by functional least squares , 1982, Journal of Applied Probability.

[8]  X. Fernique.  Regularite des trajectoires des fonctions aleatoires gaussiennes , 1975 .

[9]  C. R. Heathcote,et al.  On the estimation of slope and the identification of outliers in linear regression , 1981 .

[10]  S. Csőrgő.  Limit Behaviour of the Empirical Characteristic Function , 1981 .

[11]  Michael B. Marcus,et al.  Weak Convergence of the Empirical Characteristic Function , 1981 .

[12]  S. Csörgo.  The theory of functional least squares , 1983, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[13]  P. V. Rao,et al.  Tests of symmetry based on watson statistics , 1981 .

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

[15]  A. Feuerverger,et al.  The Empirical Characteristic Function and Its Applications , 1977 .

[16]  Kjell A. Doksum,et al.  Measures of Location and Asymmetry , 1975 .