Abstract

We develop a distribution-free, nonparametric family of optional naive

tests for high-dimensional covariance identity testing.

Within the naive test

framework of Hu and Bai (2016), we consider an operator-indexed family of

optional naive tests defined by a positive-definite operator W acting on the covariance structure.

Classical U-statistic-based identity tests, such as that of

Chen et al. (2010), arise as isotropic choices of W, thereby embedding standard

Frobenius-norm-type procedures as special cases within a single operator-based

framework. We establish high-dimensional limiting distributions under the null

and alternatives and, to our knowledge, provide the first rigorous formalization

of a locally most powerful naive test (LMPNT) for high-dimensional covariance

identity testing. We propose a spectral optimization criterion for selecting W

by aligning its eigenvectors and eigenvalues with the departure spectrum; at the

oracle level, it yields an explicit operator WΣ1p,opt that is LMPNT over a structured class of alternatives. Extensive Monte Carlo experiments under Gaussian

https://orcid.org/0000-0002-5300-5513

and non-Gaussian designs show accurate size control and substantial power gains

over existing methods.

Key words and phrases: High-dimensional covariance identity testing, Locally most powerful naive test, Optional naive test, Spectral optimization, U-statistic

Information

Preprint No.SS-2026-0001
Manuscript IDSS-2026-0001
Complete AuthorsXiaona Lu, Jiang Hu, Zhidong Bai, Haiyan Song
Corresponding AuthorsJiang Hu
Emailshuj156@nenu.edu.cn

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Acknowledgments

Zhidong Bai was supported by the Jilin Province Science and Technology

Development Plan Project (No.YDZJ202501ZYTS593).

Supplementary Materials

The Supplementary Material contains detailed proofs, additional simulation

studies, and further details for the real-data analysis.

Supplementary materials are available for download.