Abstract
The Bayes, generalized ridge mixed, and minimax estimators have similar mathematical forms. This chapter describes the conditions for these estimators to be equivalent. C.R. Rao obtained different forms of the Bayes estimator when X and F have full rank. These forms are useful for various reasons: (1) ridge type estimators may be derived as special cases of Bayes estimators, (2) the relationship between the mixed estimator and the Bayes estimator can be studied, (3) the relative weight of sample and prior information can be considered, and (4) James–Stein type estimators can be derived as special cases of empirical Bayes estimators. The chapter highlights the conditions when the generalized ridge regression estimator is equivalent to the Bayes estimator . It discusses the conditions for sample and prior information to be exchangeable. The chapter presents a generalization of the Gauss–Markov theorem. It also presents an extension of the Gauss–Markov theorem.
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