On new omnibus tests of uniformity on the hypersphere

Two new omnibus tests of uniformity for data on the hypersphere are proposed. The new test statistics exploit closed-form expressions for orthogonal polynomials, feature tuning parameters, and are related to a "smooth maximum" function and the Poisson kernel. We obtain exact moments of the test statistics under uniformity and rotationally symmetric alternatives, and give their null asymptotic distributions. We consider approximate oracle tuning parameters that maximize the power of the tests against known generic alternatives and provide tests that estimate oracle parameters through cross-validated procedures while maintaining the significance level. Numerical experiments explore the effectiveness of null asymptotic distributions and the accuracy of inexpensive approximations of exact null distributions. A simulation study compares the powers of the new tests with other tests of the Sobolev class, showing the benefits of the former. The proposed tests are applied to the study of the (seemingly uniform) nursing times of wild polar bears.