Nonparametric empirical Bayes for the Dirichlet process mixture model

The Dirichlet process prior allows flexible nonparametric mixture modeling. The number of mixture components is not specified in advance and can grow as new data arrive. However, analyses based on the Dirichlet process prior are sensitive to the choice of the parameters, including an infinite-dimensional distributional parameter G ₀. Most previous applications have either fixed G ₀ as a member of a parametric family or treated G ₀ in a Bayesian fashion, using parametric prior specifications. In contrast, we have developed an adaptive nonparametric method for constructing smooth estimates of G ₀. We combine this method with a technique for estimating α, the other Dirichlet process parameter, that is inspired by an existing characterization of its maximum-likelihood estimator. Together, these estimation procedures yield a flexible empirical Bayes treatment of Dirichlet process mixtures. Such a treatment is useful in situations where smooth point estimates of G ₀ are of intrinsic interest, or where the structure of G ₀ cannot be conveniently modeled with the usual parametric prior families. Analysis of simulated and real-world datasets illustrates the robustness of this approach.     

On a Generalization of the Classical Random Sampling Scheme and Unbiased Estimation

A generalization of the classical random sampling scheme is suggested. Based on the proposed generalization one can derive many new minimum variance unbiased estimators for probabilities, as well as for other functions of unknown parameters, for the multivariate Pólya, the multivariate negative Pólya, the multinomial, the multivariate hypergeometric, the multivariate Poisson, and the Wishart probability distributions.     

Admissible estimators in finite population sampling employing various types of prior information

We consider some estimators of the total and variance of a finite population from Bayesian and pseudo-Bayesian perspectives. Recently, Meeden and Ghosh (1982a, 1982b) have provided quite simple but powerful tools for proving admissibility of estimators and estimator-design pairs is finite population sampling problems. We consider what these techniques yield in the way of admissibility results for the estimators discussed.     

Asymptotic properties of a generalized regression-type predictor of a finite population variance in probability sampling

A system of predictors for estimating a finite population variance is defined and shown to be asymptotically design-unbiased (ADU) and asymptotically design-consistent (ADC) under probability sampling. An asymptotic mean squared error (MSE) of a generalized regression-type predictor, generated from the system, is obtained. The suggested predictor attains the minimum expected variance of any design-unbiased estimator when the superpopulation model is correct. The generalized regression-type predictor and the predictor suggested by Mukhopadhyay (1990) are compared.