On the distribution-freeness of a test of angular symmetry based on halfspace depth

Abstract

The problem of testing the null hypothesis of angular symmetry about a specified location in $R^d$ is considered, with the focus being on a well-known test based on halfspace depth. In the bivariate case $d=2$, the exact null distribution of the corresponding test statistic is explicitly known and turns out not to depend on the underlying angularly symmetric distribution, so that the test is distribution-free under the null hypothesis. Distribution-freeness, which is of course crucial to make this test applicable in practice, is further investigated here. In dimension $d=2$, the reason why distribution-freeness holds is explained and it is shown through Monte Carlo exercises that distribution-freeness does not hold in dimension $d=3$. Through suitable concepts of hyperplane arrangements, it is then investigated why the test behaves differently for $d=2$ and $d\ge 3$. The results reveal why distribution-freeness fails, and, for a particular sample size considered to ease the presentation, they show that deviations with respect to distribution-freeness, still for $d=3$, will actually remain very small. That these deviations will remain very small for other sample sizes is supported by a broader Monte Carlo exercise. Finally, a feasible conditional version of the test is proposed and asymptotic distribution-freeness is briefly discussed.