Statistics Papers

Document Type

Journal Article

Date of this Version

12-2013

Publication Source

Bernouli

Volume

19

Issue

5B

Start Page

2359

Last Page

2388

DOI

10.3150/12-BEJ455

Abstract

This paper considers testing a covariance matrix Σ in the high dimensional setting where the dimension p can be comparable or much larger than the sample size n. The problem of testing the hypothesis H0:Σ=Σ0 for a given covariance matrix Σ0 is studied from a minimax point of view. We first characterize the boundary that separates the testable region from the non-testable region by the Frobenius norm when the ratio between the dimension p over the sample size n is bounded. A test based on a U-statistic is introduced and is shown to be rate optimal over this asymptotic regime. Furthermore, it is shown that the power of this test uniformly dominates that of the corrected likelihood ratio test (CLRT) over the entire asymptotic regime under which the CLRT is applicable. The power of the U-statistic based test is also analyzed when p/n is unbounded.

Copyright/Permission Statement

The original published work is available at: https://projecteuclid.org/euclid.bj/1386078606#abstract

Keywords

correlation matrix, covariance matrix, high-dimensional data, likelihood ratio test, minimax hypothesis testing, power, testing covariance structure

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Date Posted: 27 November 2017

This document has been peer reviewed.