LINEAR BAYES AND OPTIMAL ESTIMATION

V. P. GODAMBE

Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

(Received August 26, 1996; revised September 8, 1997)

Abstract. In non-Bayesian statistics, it is often realistic to replace a full distributional assumption by a much weaker assumption about its first few moments; such as for instance, mean and variance. Along the same lines in Bayesian statistics one may wish to replace a completely specified prior distribution by an assumption about just a few moments of the distribution. To deal with such Bayesian semi-parametric models defined only by a few moments, Hartigan (1969, J. Roy. Statist. Soc. Ser. B, 31, 440–454) put forward linear Bayes methodology. By now it has become a standard tool in Bayesian analysis. In this paper we formulate an alternative methodology based on the theory of optimum estimating functions. This alternative methodology is shown to be more readily applicable and efficient in common problems, than the linear Bayes methodology mentioned above.

Key words and phrases: Bayes methodology, conditioning, estimating functions, linearity, optimality.