Bayesian estimation in the symmetric location problem

Abstract

Let Xi=θ+e{open}i for i=1, ..., n, where the e{open}i's are i.i.d. ∼F and F is symmetric about 0. F is assumed unknown or only partially known, and the problem is to estimate θ. Priors are put on the pair (F,θ). The priors on F are obtained from Doksum's neutral to the right priors, and include "symmetrized Dirichlet" priors. The marginal posterior distribution of θ given X1, ..., Xnis computed and its general properties studied. It is found that for certain classes of distributions of the e{open}i's, the posterior distribution of θ is for all large n a point mass at the true value of θ. If the distribution of the e{open}i's is not exactly symmetric, the Bayes estimates can behave very poorly. © 1984 Springer-Verlag.