On Stein's test of uniformity on the hypersphere

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 related parametric tests. Additional numerical experiments compare the proposed test with competing Sobolev-class procedures, highlighting settings in which it offers clear advantages.

Figures (5)

• Figure 1: Relative coefficients $k\mapsto c_{k,p}(\lambda)$ and $k\mapsto c^{\mathrm{dKSD}}_{k,p}(\lambda)$ for dimensions $p=3$ and $p=10$, for the $L^2$-Stein test (solid lines) and the dKSD test (dashed lines). For each choice of $\lambda$, the coefficients are standardized by their maximum. To illustrate the effect, we plot the continuous mappings in $k$, while in the test, only evaluations at integer values of $k$ are used.
• Figure 2: Hammer projection representation of $\boldsymbol{s}\mapsto \sqrt{n} |z(\boldsymbol{s})|$, for the fixed alternative $\mathrm{vMF}(\boldsymbol{\mu},\kappa)$ and $n=100$. The central point is $\boldsymbol{\mu}=(0,-1,0)^\top$. In the first row, $\kappa=1$ is fixed, while in the second, $\lambda=1$ is.
• Figure 3: Hammer projection representation of the null correlation kernel $\boldsymbol{s}\mapsto \rho(\boldsymbol{s},\boldsymbol{t})$, for $\boldsymbol{t}=(0,0,1)^\top$ (north pole, diamond). The shape of the kernel is invariant from the choice of $\boldsymbol{t}$.
• Figure 4: Hammer projection representation of the fixed-alternative correlation kernel $\boldsymbol{s}\mapsto \rho'(\boldsymbol{s},\boldsymbol{t})$, for the fixed alternative $\mathrm{vMF}(\boldsymbol{\mu},\kappa)$ and $\boldsymbol{t}=(0, 0, 1)^\top$ (north pole, diamond). The central point is $\boldsymbol{\mu}=(0,-1,0)^\top$.
• Figure 5: Powers of Stein, softmax tests, and dKSD test, with concentration parameter $\lambda$ in each column under the alternative distributions $\mathrm{MvMF}(10)$, $\mathrm{SCM}(3)$, and $\mathrm{W}(2)$. In the top row, the simulation was carried out in dimension $p=3$ while the bottom row was carried out in dimension $p=5$. Here, we use significance level $\alpha=5\%$, sample size $n=50$, and $M=1,\!000$ samples.

Theorems & Definitions (21)

• Proposition 1.1
• Remark 2.1
• Lemma 2.1
• Theorem 3.1
• Theorem 3.2
• Lemma 3.1
• Theorem 3.3
• Remark 3.1
• Remark 3.2
• Theorem 3.4