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

The problem of testing the null hypothesis of angular symmetry about a specified location in Rd 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≥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.