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

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