  The main theme of this paper is a modification of the likelihood ratio test
(LRT) for testing high dimensional covariance matrix. Recently, the correct
asymptotic distribution of the LRT for a large-dimensional case (the case $p/n$
approaches to a constant $\gamma \in (0,1]$) is specified by researchers. The
correct procedure is named as corrected LRT. Despite of its correction, the
corrected LRT is a function of sample eigenvalues that are suffered from
redundant variability from high dimensionality and, subsequently, still does
not have full power in differentiating hypotheses on the covariance matrix. In
this paper, motivated by the successes of a linearly shrunken covariance matrix
estimator (simply shrinkage estimator) in various applications, we propose a
regularized LRT that uses, in defining the LRT, the shrinkage estimator instead
of the sample covariance matrix. We compute the asymptotic distribution of the
regularized LRT, when the true covariance matrix is the identity matrix and a
spiked covariance matrix. The obtained asymptotic results have applications in
testing various hypotheses on the covariance matrix. Here, we apply them to
testing the identity of the true covariance matrix, which is a long standing
problem in the literature, and show that the regularized LRT outperforms the
corrected LRT, which is its non-regularized counterpart. In addition, we
compare the power of the regularized LRT to those of recent non-likelihood
based procedures.
