On Stein's test of uniformity on the hypersphere
Paul Axmann, Bruno Ebner, Eduardo Garc\'ia-Portugu\'es

TL;DR
This paper introduces a new Stein-based uniformity test on the hypersphere utilizing harmonic decomposition and Sobolev tests, with closed-form null distribution and strategies for tuning parameters to improve power.
Contribution
It develops a novel Stein characterization for hypersphere uniformity, deriving explicit distributions and tuning methods, and compares its performance with existing Sobolev tests.
Findings
Closed-form null distribution of the test statistic.
Tuning parameter enhances test power against alternatives.
Outperforms related Sobolev tests in specific scenarios.
Abstract
We propose a new test of uniformity on the hypersphere based on a Stein characterization associated with the Laplace--Beltrami operator. We identify a sufficient class of test functions for this characterization, linked to the moment generating function. Exploiting the operator's eigenfunctions to obtain a harmonic decomposition in terms of Gegenbauer polynomials, we show that the proposed procedure belongs to the class of Sobolev tests. We derive closed-form expressions for the distribution of the test statistic under the null hypothesis and under fixed alternatives. To enhance power against a range of alternatives, we introduce a tuning parameter into the characterization and study its impact on rejection probabilities. We discuss data-driven strategies for selecting this parameter to maximize rejection rates for a given alternative and compare the resulting performance with that of…
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Taxonomy
TopicsMarkov Chains and Monte Carlo Methods · Random Matrices and Applications · Numerical methods in inverse problems
