PREDICTIVE DENSITY ESTIMATION FOR MULTIPLE REGRESSION

Edward . George and Xinyi Xu

Econometric Theory, 2008, vol. 24, issue 02, pages 528-544

Abstract: Suppose we observe X Nm(A , 2I) and would like to estimate the predictive density p(y ) of a future Y Nn(B , 2I). Evaluating predictive estimates by Kullback Leibler loss, we develop and evaluate Bayes procedures for this problem. We obtain general sufficient conditions for minimaxity and dominance of the noninformative uniform prior Bayes procedure. We extend these results to situations where only a subset of the predictors in A is thought to be potentially irrelevant. We then consider the more realistic situation where there is model uncertainty and this subset is unknown. For this situation we develop multiple shrinkage predictive estimators and obtain general minimaxity and dominance conditions. Finally, we provide an explicit example of a minimax multiple shrinkage predictive estimator based on scaled harmonic priors.We acknowledge Larry Brown, Feng Liang, Linda Zhao, and three referees for their helpful suggestions. This work was supported by various NSF grants, DMS-0605102 the most recent.

Date: 2008

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