Rank Order Estimation with the Dirichlet Prior

Abstract

Suppose that a sample of size $n$ from a distribution function $F$ is obtained. However, only $r(< n)$ values from the sample are observed, say $X_1,\cdots, X_r$. Without loss of generality, we can consider $X_1,\cdots, X_r$ to be the first $r$ values in the (unordered) sample. The problem is to estimate the rank order $G$ of $X_1$ among $X_1,\cdots, X_n$. The situations of interest include $F$ nonrandom, either known or unknown, and $F$ random. The random case assumes that $F$ is a random distribution function chosen according to Ferguson's (Ann. Statist. 1 (1973) 209-230) Dirichlet process prior. Since this random distribution function is discrete with probability one, average ranks are used to resolve ties. A Bayes estimator (squared-error loss) of $G$ is developed for the random model. For the nonrandom distribution function model, optimal non-Bayesian estimators are developed in both the case where $F$ is known and the case where $F$ is unknown. These estimators are compared with the Dirichlet estimator on the basis of average mean square errors under both the random and nonrandom models.