The Annals of Statistics

Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size

Olivier Ledoit and Michael Wolf

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This paper analyzes whether standard covariance matrix tests work when dimensionality is large, and in particular larger than sample size. In the latter case, the singularity of the sample covariance matrix makes likelihood ratio tests degenerate, but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well-defined. We study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together, with their ratio converging to a finite nonzero limit. We find that the existing test for sphericity is robust against high dimensionality, but not the test for equality of the covariance matrix to a given matrix. For the latter test, we develop a new correction to the existing test statistic that makes it robust against high dimensionality.

Article information

Ann. Statist., Volume 30, Number 4 (2002), 1081-1102.

First available in Project Euclid: 10 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H15: Hypothesis testing
Secondary: 62E20: Asymptotic distribution theory

Concentration asymptotics equality test sphericity test


Ledoit, Olivier; Wolf, Michael. Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size. Ann. Statist. 30 (2002), no. 4, 1081--1102. doi:10.1214/aos/1031689018.

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