Metrika (2010) 72:21–35 DOI 10.1007/s00184-009-0238-3 Necessary conditions for admissibility of matrix linear estimators in a multivariate linear model Etsuo Miyaoka · Kazuo Noda Received: 22 January 2008 / Published online: 6 March 2009 © Springer-Verlag 2009 Abstract This article provides necessary conditions for the admissibility of matrix linear estimators of an estimable parameter matrix linear function under two kinds of quadratic matrix loss functions in a multivariate linear model following a family of matrix normal distributions, where the covariance matrix associated is completely unknown. Further it is demonstrated that if a more concrete condition supplied for one of the subdivided conditions is satisfied, then the special condition concerning the Stein problem is necessary for the admissibility of the kind of estimators under each of the loss functions. Keywords Parameter matrix linear function · Quadratic matrix loss functions · Matrix normal distributions · Unknown covariance matrix · The Stein problem · James-Stein type matrix estimator Mathematics Subject Classification (2000) Primary 62C15; Secondary 62H12 1 Introduction Let us consider a multivariate linear model, Y = XB + ε (1.1) where Y = (y 1 , y 2 ,..., y n ) ′ is an n × m matrix of observable random variables, X is a known n × p design matrix, B is an unknown p × m parameter matrix, ε = (e 1 , e 2 ,..., e n ) ′ is an n × m error matrix having a matrix normal distribution, E. Miyaoka (B ) · K. Noda Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, Japan e-mail: miyaoka@rs.kagu.tus.ac.jp 123