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Abstract
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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
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