On the Properties of Estimates of Monotonic Mean Vectors for Multivariate Normal Distributions

Problems concerning estimation of parameters and determination the statistic, when it is known a priori that some of these parameters are subject to certain order restrictions, are of considerable interest. In the present paper, we consider the estimators of the monotonic mean vectors for two dimensional normal distributions and compare those with the unrestricted maximum likelihood estimators under two different cases. One case is that covariance matrices are known, the other one is that covariance matrices are completely unknown and unequal. We show that when the covariance matrices are known, under the squared error loss function which is similar to the mahalanobis distance, the obtained multivariate isotonic regression estimators, motivated by estimators given in Robertson et al. (1988), which are the estimators given by Sasabuchi et al. (1983) and Sasabuchi et al. (1992), have the smaller risk than the unrestricted maximum likelihood estimators uniformly, but when the covariance matrices are unknown and unequal, the estimators have the smaller risk than the unrestricted maximum likelihood estimators only over some special sets which are defined on the covariance matrices. To illustrate the results two numerical examples are presented.