Bayesian Nonparametric Estimation of the Median; Part II: Asymptotic Properties of the Estimates

Abstract

For data $\theta + \varepsilon_i, i = 1, \ldots, n$ where $\varepsilon_i$ are i.i.d. $\sim F$ with the median of $F$ equal to $0$ but $F$ otherwise unknown, it is desired to estimate $\theta$. In Doss (1985) priors are put on the pair $(F, \theta)$, the marginal posterior distribution of $\theta$ is computed, and the mean of the posterior is taken as the estimate of $\theta$. In the present paper a frequentist point of view is adopted. The consistency properties of the Bayes estimates computed in Doss (1985) are investigated when the prior on $F$ is of the "Dirichlet-type." Any $F$ whose median is 0 is in the support of these priors. It is shown that if the $\varepsilon_i$ are i.i.d. from a discrete distribution, then the Bayes estimates are consistent. However, if the distribution of the $\varepsilon_is$ is continuous, the Bayes estimates can be inconsistent.