Testing symmetry about an unknown median, via linear rank procedures

The properties of several linear rank statistics for testing the hypothesis that a random variable with unknown median has a symmetric distribution, are explored. Each test is related, in a natural way, to a rank type estimator of the median. Specifically, the tests are based on swre functions which are the third element of a complete orthonormal basis for L 2[O,1] and the median is estimated by a natural associated R-estimator. It is shown that by linking the score function and estimator in this fashion the resulting tests have a simple asymptotic distribution under both the null hypothesis and sequences of local alternatives. Based on the asymptotic distribution theory, simple small sample modifications are presented. It is shown that for small n, even when approximate asymptotic critical values are used, these test maintain a stable a-level, which is close to the nominal level, over a wide range of symmetric distributions. Also with respect to power, the tests are competitive with existing testing proc...

[1]  E. F. Schuster.  Identifying the Closest Symmetric Distribution or Density Function , 1987 .

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

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

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

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

[6]  Simon J. Sheather,et al.  Jackknifing R-estimators , 1989 .

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

[8]  P. Sen.  Tied-Down Wiener Process Approximations for Aligned Rank Order Processes and Some Applications , 1977 .

[9]  P. Sen,et al.  On sequentially adaptive signed-rank statistics , 1986 .

[10]  Götz Kersting,et al.  Tests for symmetry , 1977 .

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

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

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

[14]  R. Rosenstein.  Tests of symmetry derived as components of pearson's phi-squared distance measure , 1989 .

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

[16]  Thomas P. Hettmansperger,et al.  12 On the use of rank tests and estimates in the linear model , 1984, Nonparametric Methods.

[17]  Tore Schweder,et al.  Window Estimation of the Asymptotic Variance of Rank Estimators of Location. , 1973 .

[18]  A note on testing symmetry with estimated parameters , 1985 .

[19]  A test for skewness with ordered variables. , 1954, Annals of eugenics.

[20]  Husˇkova´ Marie,et al.  On sequentially adaptive asymptotically efficient rank statistics , 1985 .