Bayesian Nonparametric Estimation of the Median; Part I: Computation of the Estimates

The Annals of Statistics · 1985 · Volume 13 · Issue 4

Abstract

Let $X_i, i = 1, \ldots, n$ be i.i.d. $\sim F_\theta$, where $F_\theta(x) = F(x - \theta)$ for some $F$ that has median equal to 0. $F$ is assumed unknown or only partially known, and the problem is to estimate $\theta$. Priors are put on the pair $(F, \theta)$. The priors on $F$ concentrate all their mass on c.d.f.s with median equal to 0. These priors include "Dirichlet-type" priors. The marginal posterior distribution of $\theta$ given $X_1, \ldots, X_n$ is computed. The mean of the posterior is taken as the estimate of $\theta$.

Keywords

62F15, 62G05, Bayes estimator, Dirichlet process priors, estimation of quantiles, estimation of the median, regular conditional distribution

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