APPROXIMATE BAYESIAN SHRINKAGE ESTIMATION

WANG-SHU LU

Department of Statistics, College of Arts and Science, University of Missouri-Columbia, 222 Math Sciences Bldg., Columbia, MO 65211, U.S.A.

(Received March 8, 1993; revised February 8, 1994)

Abstract. A Bayesian shrinkage estimate for the mean in the generalized linear empirical Bayes model is proposed. The posterior mean under the empirical Bayes model has a shrinkage pattern. The shrinkage factor is estimated by using a Bayesian method with the regression coefficients to be fixed at the maximum extended quasi-likelihood estimates. This approach develops a Bayesian shrinkage estimate of the mean which is numerically quite tractable. The method is illustrated with a data set, and the estimate is compared with an earlier one based on an empirical Bayes method. In a special case of the homogeneous model with exchangeable priors, the performance of the Bayesian estimate is illustrated by computer simulations. The simulation result shows as improvement of the Bayesian estimate over the empirical Bayes estimate in some situations.

Key words and phrases: Bayes estimate, empirical Bayes, extended quasilikelihood, generalized linear model, relative saving loss, shrinkage estimate.