A class of estimators dominating the james-stein estimator shrinking toward projection vectors under the balanced loss function

In this paper, we consider the problem of estimating the mean vector of a multi-variate normal distribution with an unknown diagonal covariance matrix. Two classes of estimators for the mean vector are proposed and their performances are evaluated under the balanced loss function. First, we introduce a class of estimators derived from the maximum likelihood estimator (MLE) and establish a sufficient condition on the shrinkage function under which these estimators uniformly improve upon the MLE in terms of risk. Next, by combining the MLE with the James-Stein estimator that shrinks toward projection vectors, we construct a new class of estimators. Under a simple and practically verifiable condition, we show that the proposed estimators dominate the James-Stein estimator shrinking toward projection vectors with respect to the risk function, thereby explaining their superior performance. Finally, numerical studies are presented to illustrate and confirm the effectiveness of the proposed estimators.