Inadmissibility of the maximum likelihood estimator for a multivariate normal distribution when some observations are missing

Abstract

The random variables have a (p+1)-variate normal distribution. Suppose we observe m independent observations on the (p+L)- dimensional random vector . We assume that n additional, independent observations on Xare available hut the corresponding n observations on Y are missing. Anderson (1957) considered the problem of estimating the mean of the marginal distribution of Y and derived the maximum likelihood estimator (MLE) for the mean. In this paper it is shown that the MLE is inadmissible with respect to squared error loss function. The problem of finding a certain class of estimators which have everywhere smaller risk than the MLE for estimating the mean of Y is reduced to the problem of finding a set of estimators which have everywhere smaller risk than the MLE for the regression parameter vector --- of Y on X. Estimators for the mean of Y based oon Baranchik's estimators (1973) for β are obtained and it is shown that they strictly dominate the.MLE. It is also shown that Baranchik's estimator for β can ...