Paper #575

Title:

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

Authors:

Olivier Ledoit and Michael Wolf

Date:

October 2001

Abstract:

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 non-zero 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.

Keywords:

Concentration asymptotics, equality test, sphericity test

JEL codes:

C12, C52

Area of Research:

Statistics, Econometrics and Quantitative Methods

Published in:

Annals of Statistics 30, 1081-1102, 2002

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