## The Annals of Statistics

### High-dimensional covariance matrices in elliptical distributions with application to spherical test

#### Abstract

This paper discusses fluctuations of linear spectral statistics of high-dimensional sample covariance matrices when the underlying population follows an elliptical distribution. Such population often possesses high order correlations among their coordinates, which have great impact on the asymptotic behaviors of linear spectral statistics. Taking such kind of dependency into consideration, we establish a new central limit theorem for the linear spectral statistics in this paper for a class of elliptical populations. This general theoretical result has wide applications and, as an example, it is then applied to test the sphericity of elliptical populations.

#### Article information

Source
Ann. Statist., Volume 47, Number 1 (2019), 527-555.

Dates
Revised: October 2017
First available in Project Euclid: 30 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1543568597

Digital Object Identifier
doi:10.1214/18-AOS1699

Mathematical Reviews number (MathSciNet)
MR3909941

Zentralblatt MATH identifier
07036210

Subjects
Primary: 62H15: Hypothesis testing
Secondary: 62H10: Distribution of statistics

#### Citation

Hu, Jiang; Li, Weiming; Liu, Zhi; Zhou, Wang. High-dimensional covariance matrices in elliptical distributions with application to spherical test. Ann. Statist. 47 (2019), no. 1, 527--555. doi:10.1214/18-AOS1699. https://projecteuclid.org/euclid.aos/1543568597

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#### Supplemental materials

• Supplement to “High-dimensional covariance matrices in elliptical distributions with application to spherical test”. This supplementary material gives a general result for the CLT of the moments of sample eigenvalues, proofs of Theorem 3.3 and Lemmas 2.1, 5.1, A.1–A.4, and additional simulations for assessing the tests $T_{1}$, $T_{2}$ and $T_{m}$.