Comparing Bayesian and frequentist estimators in the exchangeable case

Abstract

Consider a sample of observations (X1,...Xn) with density fQ(X1,...,Xn)= [integral operator][Theta]Pi=1nf[Theta](Xi)dQ([Theta]), where f[theta](x) is a known model, [theta] [set membership, variant] [Theta], a finite-dimensional space, and Q is an unknown distribution ranging over a suitable set of probability distributions over [Theta]. This is the most interesting case of exchangeable observations. The problem of estimating h(Q), a real-valued function of Q of interest, is considered, both from a Bayesian and a frequentist perspective. In particular, it is proved that the uniformly minimum variance unbiased estimator (UMVUE) for h([theta]) is the UMVUE for h(Q) = EQ[h([theta])]. Finally, following the idea in the paper by Samaniego and Reneau (1994), Bayesian and frequentist estimators of h(Q) are compared.Exchangeable observations Bayesian estimators Uniformly minimum variance unbiased estimators Mean squared error