Minimax estimators in the normal MANOVA model

Robust Improvement in Estimation of a Mean Matrix in an Elliptically Contoured Distribution

“…Corollary 2.1 implies that these estimators $ EM (c 1 , c 2 ) and $ KN V have robust improvements over the crude Efron Morris estimator $ EM (c 1 , 0), being better than the least squares estimator x. Similarly, other classes of estimators with respect to both kinds of loss functions given by Bilodeau and Kariya (1989) are also robust from our Theorem 2.1 and Corollary 2.1.…”

“…In this paper, we provide a complete analogue of the results obtained by Bilodeau and Kariya (1989) and Konno (1991Konno ( , 1992 for the ECD model. We also extend the results of Gleser (1987) and Honda (1991) for noninvariant loss functions.…”

“…In this general setup, Bilodeau and Kariya (1989) derived a condition for $(G) given by (2.7) to dominate x through an unbiased estimator of the risk matrix R 1 ($(G), (%, 7), f ) in the normal distribution. Our objective is to extend their general results to the ECD model.…”

“…Under the same assumptions as in Theorem 2.1, the R 2 -risk difference of the estimators x and $(G) is evaluated as Bilodeau and Kariya (1989) derived several classes of improved estimators under very general conditions. In particular, it includes Konno's (1992) class of minimax estimators, which are invariant under the group of transformations (x, S) Ä (OxC, C t SC) where O is an m_m orthogonal matrix and C is a p_p nonsingular matrix.…”

“…In Section 2, we consider the estimation of the coefficient matrix ; in the ECD model relative to invariant loss and, based on the extended Stein and Stein Haff identities, we show that all minimax (or improved) estimators derived through the general result of Bilodeau and Kariya (1989) remain robust under the ECD model. The robustness of improvement for noninvariant loss is established in Section 3 and an extension to a growth curve model is provided in Section 4.…”