# Some Hypothesis Tests for the Covariance Matrix When the Dimension Is Large
Compared to the Sample Size

## Olivier Ledoit and Michael Wolf

### 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.

Annals of
Statistics, Volume 30, Number 4, August
2002, pages 1081-1102

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