On the robustness to symmetry of some nonparametric multivariate one-sample sign-type tests

Many nonparametric multivariate one-sample tests in the literature require the assumption of symmetry or directional symmetry to be distribution-free. Examples include the bivariate sign test of Blumen [A new bivariate sign test. J Am Stat Assoc. 1958;53(282):448–456], the bivariate sign test of Brown and Hettmansperger [An affine invariant bivariate version of the sign Test. J R Stat Soc B. 1989;51(1):117–125], the multivariate sign test of Randles [A simpler, affine invariant, multivariate, distribution-free sign test. J Am Stat Assoc. 2000;95:1263–1268] and the multivariate signed-rank test of Oja and Randles [Multivariate nonparametric tests. Stat Sci. 2004;19:598–605]. In the current literature, it is not known how robust these tests are to the assumption of (directional) symmetry. When the symmetry assumption is not satisfied, the observed or attained significance level (the type I error probability or the size) of these tests may be unacceptably different from the specified nominal value. In this paper, we examine this robustness issue for the four nonparametric multivariate one-sample sign-type tests cited above in a simulation study, using data from several families of distributions. The popular parametric Hotelling's $T2$ test included as a benchmark. Conclusions and recommendations are offered.