Addressing parity blindness of data-driven Sobolev tests on the hypersphere

TL;DR

This paper analyzes the limitations of data-driven Sobolev tests for uniformity on hyperspheres, identifies their blindness to certain alternatives, and proposes a modification that improves their power while maintaining consistency.

Contribution

It introduces a simple modification to the Sobolev test that enhances its ability to detect contiguous alternatives, addressing parity blindness.

Findings

Modified test retains consistency under fixed alternatives.

Enhanced power against certain contiguous alternatives.

Simulation results confirm theoretical improvements.

Abstract

We study the asymptotic behavior of the data-driven Sobolev test for testing uniformity on the (hyper)sphere. We show that it can be blind to certain contiguous alternatives and propose a simple modification of the test statistic. This adapted test retains consistency under fixed alternatives and achieves non-trivial asymptotic power against contiguous alternatives for which the original test fails. Simulation results support our theoretical findings.