Comparing Bayesian and frequentist estimators in the exchangeable case

Abstract

Consider a sample of observations ( X 1,… X n ) with density f Q(X 1,…,X n)= ∫ ϴП i=1 nfϴ(X i)dQ(ϴ) , where f θ ( x) is a known model, θ ∈ Θ, a finite-dimensional space, and Q is an unknown distribution ranging over a suitable set of probability distributions over Θ. 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( θ) is the UMVUE for h( Q) = E Q [ h( θ)]. Finally, following the idea in the paper by Samaniego and Reneau (1994), Bayesian and frequentist estimators of h( Q) are compared.