This paper addresses the Stein conjecture in the simultaneous estimation of a matrix mean
of a multivariate normal distribution with a known covariance matrix. Stein (1973) derived an
unbiased estimator of a risk function for orthogonally equivariant estimators and considered to
isotonize the estimator which minimizes the main part of the unbiased risk-estimator. We call it
the Stein risk-minimization estimator (RM) in this paper. Although the Stein RM estimator has
been recognized as an excellent procedure with a nice risk-performance, it has a complicated
form based on the isotonizing algorithm, and no analytical properties such as minimaxity have
been shown. The aim of this paper is to fix this conjecture in lower dimensional cases, that is,
the minimaxity of the Stein RM estimator is established for the two and three dimensions.
|