Research on Shrinkage Estimator

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The exact risk function of the generalized Bayes estimator for bivariate normalcovariance matrix against reference prior due to Yang and Berger(1994, Ann. Statist.) is derived. It is compared with those of the Stein's minimax estimator, shrinkage estimator, Haff's estimator and orthogonally equivariant estimator by Takemura. It is found that the generalized Bayes estimator does not dominate maximum likelihood estimator at some region of the ratio of the latent root far from one and that it is very good near one.The formula for calculating the exact power of classically famous four criteria ; likelihood ratio, maximum latent root, Hotelling's trace and Pillai's locally best invariant test for bivariate linear hypotheses are derived and the graphs of the power surfaces are demonstrated to see the global behavior. It is found that the ordering of the power : Hotelling's test<Likelihood ratio test<Pillai's test at the neighborhood of the null hypotheses and the reversed inequalities at far from the null hypotheses do not necessarily hold, though they do hold for large sample sizes.The generalized Bayes test for loop-ordered normal means is derived which has almost the same minimum power for given non-centrality and has much larger maximum power than likelihood ratio test. It is found that the non-informative prior should be taken at the most distant corner vectors of the polyhedral convex cone formed by the alternative hypotheses. It is shown that the most distant corner vectors are made from pairs of two, each of which intersect the center of the cone with equal angle.