High-dimensional Sobolev tests on hyperspheres

TL;DR

This paper derives the asymptotic null distribution of Sobolev tests for uniformity on high-dimensional hyperspheres, analyzing their behavior under local alternatives and applying results to test for rotational and spherical symmetry.

Contribution

It provides the first comprehensive asymptotic analysis of Sobolev tests in high dimensions with diverging sample size and dimension, including their behavior under local alternatives.

Findings

Derived the limit null distribution for Sobolev tests in high dimensions.

Characterized the tests' behavior under local von Mises-Fisher alternatives.

Validated results through numerical experiments under various hypotheses.

Abstract

We derive the limit null distribution of the class of Sobolev tests of uniformity on the hypersphere when the dimension and the sample size diverge to infinity at arbitrary rates. The limiting non-null behavior of these tests is obtained for a sequence of integrated von Mises-Fisher local alternatives. The asymptotic results are applied to test for high-dimensional rotational symmetry and spherical symmetry. Numerical experiments illustrate the derived behavior of the uniformity and spherically symmetry tests under the null and under local and fixed alternatives.