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 ...
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.
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