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

Marcio Reverbel

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.

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

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.
Paper Structure (6 sections, 6 theorems, 19 equations, 1 figure)

This paper contains 6 sections, 6 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Data-driven Sobolev tests
  4. An adapted data-driven Sobolev test
  5. Simulations
  6. Conclusion

Key Result

Corollary 3.1

For $d\ge2$, consider a sequence $v_k$ with only finitely many non‐zero terms such that $v_k=0$ for each even (resp. odd) $k$. Let $g$ be an angular function, $q \;(\ge \max\{k : v_k \neq 0\})$ times differentiable at zero, with $g^{(k)}(0)=0$ for each odd (resp. even) $k\in\{k_v,\dots,q\}$. Then, u

Figures (1)

  • Figure 1: Rejection frequencies at asymptotic level $\alpha=5\%$ for $5000$ samples under (top) von Mises–Fisher and (bottom) Watson alternatives; left: data-driven Sobolev, right: adapted version. Their respective angular functions are of the form $g(\kappa_n \bm x^\top \bm \mu)$, where $\kappa_n = n^{-1/\ell}\tau$ and $\bm \mu = \bm e_1$, from the standard basis. In the bottom figures, we included in gray the asymptotic power of the Bingham test, for comparison.

Theorems & Definitions (9)

  • Corollary 3.1: Parity‐blindness
  • Lemma 4.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • Theorem 4.3
  • Corollary 4.1
  • proof