Hypothesis testing for sphericity covariance structure without a normality condition

Jieqiong Shen; Jieqiong Shen

doi:10.3934/era.2026231

Electronic Research Archive

2026, Volume 34, Issue 8: 5200-5216. doi: 10.3934/era.2026231

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Research article

Hypothesis testing for sphericity covariance structure without a normality condition

Jieqiong Shen ^,

School of Computer and Data Engineering, NingboTech University, Ningbo 315100, China

Received: 30 March 2026 Revised: 28 May 2026 Accepted: 09 June 2026 Published: 17 June 2026

This paper investigated the sphericity test for high-dimensional covariance matrices, which is a fundamental problem in probability and statistics courses. While several existing powerful tests in textbooks and literature are available, they often rely on either the normality assumption or an unknown fourth moment. This work aimed to relax these restrictions, serving as a complement and extension of traditional course content on this subject. Specifically, we constructed a consistent estimator related to the fourth moment and proposed a new test procedure. The proposed test statistic depends on the second, third, and fourth arithmetic means of the eigenvalues of the sample covariance matrix, but does not require the normality assumption. Numerical simulations illustrated that the proposed test maintains satisfactory empirical size, and achieves power comparable to, or in some cases superior to, the existing tests.
- hypothesis test,
- sphericity,
- covariance matrix,
- non-normality,
- high-dimensional data
Citation: Jieqiong Shen. Hypothesis testing for sphericity covariance structure without a normality condition[J]. Electronic Research Archive, 2026, 34(8): 5200-5216. doi: 10.3934/era.2026231

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Abstract

This paper investigated the sphericity test for high-dimensional covariance matrices, which is a fundamental problem in probability and statistics courses. While several existing powerful tests in textbooks and literature are available, they often rely on either the normality assumption or an unknown fourth moment. This work aimed to relax these restrictions, serving as a complement and extension of traditional course content on this subject. Specifically, we constructed a consistent estimator related to the fourth moment and proposed a new test procedure. The proposed test statistic depends on the second, third, and fourth arithmetic means of the eigenvalues of the sample covariance matrix, but does not require the normality assumption. Numerical simulations illustrated that the proposed test maintains satisfactory empirical size, and achieves power comparable to, or in some cases superior to, the existing tests.

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