Tests of symmetry based on the bootstrap estimation of the asymptotic variance of a skewness coefficient

Abstract

Testing for symmetry about an unknown median is a problem that has attracted the attention of statisticians for decades. Many of the existing tests of symmetry are based on the asymptotic distribution of a skewness coefficient, which typically is asymptotically normal and centered around zero for symmetric distributions. Unfortunately, the asymptotic variance depends on the underlying distribution and differs from one symmetric distribution to another one. A possible way out is to estimate this asymptotic variance from the sample by means of the bootstrap. In this paper, we explore this approach for six different skewness coefficients existing in the literature. Extensive experiments are performed for comparing the performance of the six associated tests of symmetry to that of two state-of-the-art symmetry tests in terms of preservation of the significance level under several symmetric distributions, power under asymmetric distributions and robustness in the presence of outliers. Even though the results show no clear best test, we conclude by providing some guidelines for choosing a test of symmetry based on the needs of the user.