Asymptotically Optimal Nonparametric Empirical Bayes Via Predictive Recursion

An empirical Bayes problem has an unknown prior to be estimated from data. The predictive recursion (PR) algorithm provides fast nonparametric estimation of mixing distributions and is ideally suited for empirical Bayes applications. This article presents a general notion of empirical Bayes asymptotic optimality, and it is shown that PR-based procedures satisfy this property under certain conditions. As an application, the problem of in-season prediction of baseball batting averages is considered. There the PR-based empirical Bayes rule performs well in terms of prediction error and ability to capture the distribution of the latent features.     

Optimal control designs using predicting densities for the multivariate linear model

We provide an application of a variety of predicting densities to quality control involving multivariate normal linear models. We produce optimal control designs for single muleivaiiate future observations using predicting densities employing estimative, profile likelihood, Hinkley-Lauritzen, Butler, Bayesian, and Parametric Bootstrap methodologies. The decision-theoretic optimality criterion is an intuitively appealing quadratic consumer-producer risk function. The optimal control design arising from an optimal Kullback-Leibler frequentist prediction density is shown to coincide with that arising from an optimal Kullback-Leibler Bayesian predictive density. An example involving EVOP is provided to illustrate the methodology and to raise questions concerning the relative merics of the variety of predictive approaches in the quality control context.     

Parameter Estimation Through Weighted Least-Squares Rank Regression with Specific Reference to the Weibull and Gumbel Distributions

Probability plots are often used to estimate the parameters of distributions. Using large sample properties of the empirical distribution function and order statistics, weights to stabilize the variance in order to perform weighted least squares regression are derived. Weighted least squares regression is then applied to the estimation of the parameters of the Weibull, and the Gumbel distribution. The weights are independent of the parameters of the distributions considered. Monte Carlo simulation shows that the weighted least-squares estimators outperform the usual least-squares estimators totally, especially in small samples.     

An Approximate Bayesian Marginal Likelihood Approach for Estimating Finite Mixtures

Estimation of finite mixture models when the mixing distribution support is unknown is an important problem. This article gives a new approach based on a marginal likelihood for the unknown support. Motivated by a Bayesian Dirichlet prior model, a computationally efficient stochastic approximation version of the marginal likelihood is proposed and large-sample theory is presented. By restricting the support to a finite grid, a simulated annealing method is employed to maximize the marginal likelihood and estimate the support. Real and simulated data examples show that this novel stochastic approximation and simulated annealing procedure compares favorably with existing methods.     

A review of classic non-asymptotic methods for comparing two proportions by means of independent sampLES

The 2 × 2 tables used to present the data in an experiment for comparing two proportions by means of two observations of two independent binomial distributions may appear simple but are not. The debate about the best method to use is unending, and has divided statisticians into practically irreconcilable groups. In this article, all the available non-asymptotic tests are reviewed (except the Bayesian methodology). The author states which is the optimal (for each group), referring to the tables and programs that exist for them, and contrast the arguments used by supporters of each of the options. They also sort the tangle of solutions into "families", based on the methodology used and/or prior assumptions, and point out the most frequent methodological mistakes committed when comparing the different families.     

Basu's Lemma and Goodness-of-Fit Tests

In this paper we develop a technique that is useful in goodness of fit problems with nuisance location and scale parameters. Using Basu's lemma, we transform the sample so that it is independent of the nuisance parameters. The technique will be illustrated by several examples, a discussion of the efficiency of the proposed procedure, and some concluding remarks will given.