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.
"Rank Order Estimation with the Dirichlet Prior." Ann. Statist. 6 (1) 142 - 153, January, 1978. https://doi.org/10.1214/aos/1176344073