### BRANI VIDAKOVIC

Institute of Statistics and Decision Sciences, Duke University,
Box 90251, Durham, NC 27708-0251, U.S.A.

(Received March 10, 1997; revised September 29, 1997)

Abstract.    The problem under consideration is the Gamma-minimax estimation, under L2 loss, of a multivariate normal mean when the covariance matrix is known. The family Gamma of priors is induced by mixing zero mean multivariate normals with covariance matrix tau I by nonnegative random variables tau, whose distributions belong to a suitable family \cal G. For a fixed family \cal G, the linear (affine) Gamma-minimax rule is compared with the usual Gamma-minimax rule in terms of corresponding Gamma-minimax risks. It is shown that the linear rule is good'', i.e., the ratio of the risks is close to 1, irrespective of the dimension of the model. We also generalize the above model to the case of nonidentity covariance matrices and show that independence of the dimensionality is lost in this case. Several examples illustrate the behavior of the linear Gamma-minimax rule.

Key words and phrases:    Affine rules, efficiency, Bayes risk, Brown's identity, Gamma-minimax rules.

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