Bayesian Estimation and Prediction for the Beta-Binomial Model

The beta-binomial distribution, which is generated by a simple mixture model, has been widely applied in the social, physical, and health sciences. Problems of estimation, inference, and prediction have been addressed in the past, but not in a Bayesian framework. This article develops Bayesian procedures for the beta-binomial model and, using a suitable reparameterization, establishes a conjugate-type property for a beta family of priors. The transformed parameters have interesting interpretations, especially in marketing applications, and are likely to be more stable. More specifically, one of these parameters is the market share and the other is a measure of the heterogeneity of the customer population. Analytical results are developed for the posterior and prediction quantities, although the numerical evaluation is not trivial. Since the posterior moments are more easily calculated, we also propose the use of posterior approximation using the Pearson system. A particular case (when there are two trials), which occurs in taste testing, brand choice, media exposure, and some epidemiological applications, is analyzed in detail. Simulated and real data are used to demonstrate the feasibility of the calculations. The simulation results effectively demonstrate the superiority of Bayesian estimators, particularly in small samples, even with uniform (“non-informed”) priors. Naturally, “informed” priors can give even better results. The real data on television viewing behavior are used to illustrate the prediction results. In our analysis, several problems with the maximum likelihood estimators are encountered. The superior properties and performance of the Bayesian estimators and the excellent approximation results are strong indications that our results will be potentially of high value in small sample applications of the beta-binomial and in cases in which significant prior information exists.     

An F Approximation to the Distribution of a Linear Combination of Chi-squared Variables.

A three-parameter F approximation to the distribution of a positive linear combination of central chi-squared variables is described. It is about as easy to implement as the Satterthwaite-Welsh and Hall-Buckley-Eagleson approximations. Some reassuring properties of the F approximation are derived, and numerical results are presented. The numerical results indicate that the new approximation is superior to the Satterthwaite approximation and, for some purposes, better than the Hall-Buckley-Eagleson approximation. It is not quite as good as the Gamma-Weibull approximation due to Solomon and Stephens, but is easier to implement because iterative methods are not required.     

Saddlepoint Approximations to the Density and the Distribution Functions of Linear Combinations of Ratios of Partial Sums

A class of ratios of partial sums, including Normal, Weibull, Gamma, and Exponential distributions, is considered. The distribution of a linear combination of ratios of partial sums from this class is characterized by the distribution of a linear combination of Dirichlet components. This article presents two saddlepoint approaches to calculate the density and the distribution function for such a class of linear combinations. A simulation study is conducted to assess the performance of the saddlepoint methods and shows the great accuracy of the approximations over the usual asymptotic approximation. Applications of the presented approximations in statistical inferences are discussed.     

Prediction of Times to Failure of Censored Items for a Simple Step-Stress Model with Regular and Progressive Type I Censoring from the Exponential Distribution

The problem of predicting times to failure of units from the Exponential Distribution which are censored under a simple step-stress model is considered in this article. We discuss two types of censoring—regular and progressive Type I—and two kinds of predictors—the maximum likelihood predictors (MLP) and the conditional median predictors (CMP) for each type of censoring. Numerical examples are used to illustrate the prediction methods. Using simulation studies, mean squared prediction error (MSPE) and prediction intervals are generated for these examples. MLP and the CMP are then compared with respect to MSPE and the prediction interval.     

Efficiency of a Linear Combination of the Median and the Sample Mean: The Double Truncated Normal Distribution

Abstract.  We characterize all symmetric location models for which a linear combination of the median and the sample mean is an asymptotically efficient estimator of the location parameter. The resulting model can be understood as a symmetrized or double truncated normal distribution. A simple algorithm to estimate the parameters is given and an application is presented.     

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

In this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.     

On the Built-in Restrictions in Linear Mixed Models,with Application to Smoothing Spline Analysis of Variance

The best linear unbiased predictor (BLUP) of the random parameter in a linear mixed model satisfies a linear constraint, which has been previously termed a built-in restriction. In other literature, constraints on the random parameter itself have been introduced into the modeling framework. The present article has two goals. First, it explores the idea of imposing the built-in restrictions on the BLUP as constraints on the random parameter. Second, it investigates the built-in restrictions satisfied by certain smoothing spline analysis of variance (SSANOVA) estimators, and compares these restrictions to arguably more natural side conditions on the ANOVA decomposition.     

On the information-based measure of covariance complexity and its application to the evaluation of multivariate linear models

This paper introduces a new information-theoretic measure of complexity called ICOMP as a decision rule for model selection and evaluation for multivariate linear models. The development of ICOMP is based on the generalization and utilization of the covariance complexity index of van Emden (1971) in estimation of the multivariate linear model. ICOMP is motivated by Akaike's (1973) Information Criterion (AIC), but it is a different procedure than AIC. In linear or nonlinear statistical models ICOMP uses an information-based characterization of: (i) the covariance matrix properties of the parameter estimates of a model starting from their finite sampling distributions, and (ii) the complexity of the inverse-Fisher information matrix (i-FIM) as a new criterion of achievable accuracy of the model As a result, it provides a trade-off between the accuracy of the parameter estimates and the interaction of the residuals of a model via the measure of complexity of their respective covariances. It controls the risks of both insufficient and overparameterized models, and incorporates the assumption of dependence and the independence of the residuals in one criterion function. A model with minimum ICOMP is chosen to be the best model among all possible competing alternative models. ICOMP relieves the researcher of any need to consider the parameter dimension of a model explicitly. A real numerical example is shown in subset selection of variables in multivariate regression analysis to demonstrate the utility and versatility of the new approach.