Proportionality of Covariance Matrices

Abstract

The present paper considers inference for the statistical model specifying proportionality between independent Wishart distributions. It is shown that the maximum likelihood estimate of the common covariance matrix and the constants of proportionality is the unique solution to the likelihood equation and an iterative procedure determining the estimator is given. The model is an exponential transformation model, which implies the existence of an exact ancillary (a maximal invariant), and an asymptotic expansion for the distribution of the estimator conditionally on the ancillary is given. In addition, it is shown that the estimator is unbiased to order $O(n^{-3/2})$. Finally, we derive the Bartlett adjustment to the likelihood ratio statistic for the hypothesis of proportionality and subsequently for the hypothesis specifying that all of the constants of proportionality are equal to one. A small simulation study shows that the Bartlett adjustment can be very effective in improving the accuracy of chi-squared approximations.