Bayes and Empirical Bayes Estimation of the Parameter n in a Binomial Distribution

The problem of estimating the total number of trials n in a binomial distribution is reconsidered in this article for both cases of known and unknown probability of success p from the Bayesian viewpoint. Bayes and empirical Bayes point estimates for n are proposed under the assumption of a left-truncated prior distribution for n and a beta prior distribution for p. Simulation studies are provided in this article in order to compare the proposed estimate with the most familiar n estimates.     

On the Bayesian Estimation for the Uniform Scale Parameter via a Functional Equation

Bayes uniform model under the squared error loss function is shown to be completely identifiable by the form of the Bayes estimates of the scale parameter. This results in solving a specific functional equation. A complete characterization of differentiable Bayes estimators (BE) and generalized Bayes estimators (GBE) is given as well as relations between degrees of smoothness of the estimators and the priors. Characterizations of strong (generalized Bayes) Bayes sequence (SBS or SGBS) are also investigated. A SBS is a sequence of estimators (one for each sample size) where all its components are BE generated by the same prior measure. A complete solution is given for polynomial Bayesian estimation.     

Empirical Bayes sequential estimation of a finite population mean

In this article, we consider the Bayes and empirical Bayes problem of the current population mean of a finite population when the sample data is available from other similar (m-1) finite populations. We investigate a general class of linear estimators and obtain the optimal linear Bayes estimator of the finite population mean under a squared error loss function that considered the cost of sampling. The optimal linear Bayes estimator and the sample size are obtained as a function of the parameters of the prior distribution. The corresponding empirical Bayes estimates are obtained by replacing the unknown hyperparameters with their respective consistent estimates. A Monte Carlo study is conducted to evaluate the performance of the proposed empirical Bayes procedure.     

Robust Bayesian analysis of generalized inverted family of distributions

ABSTRACT

In the current study we develop the robust Bayesian inference for the generalized inverted family of distributions (GIFD) under an ε-contamination class of prior distributions for the shape parameter α, with different possibilities of known and unknown scale parameter. We used Type II censoring and Bartholomew sampling scheme (1963) for the following derivations under the squared-error loss function (SELF) and linear exponential (LINEX) loss function : ML-II Bayes estimators of the i) parameters; ii) Reliability function and; iii) Hazard function. We also present simulation study and analysis of a real data set.     

Learning from a lot: Empirical Bayes for high‐dimensional model‐based prediction

Empirical Bayes is a versatile approach to “learn from a lot” in two ways: first, from a large number of variables and, second, from a potentially large amount of prior information, for example, stored in public repositories. We review applications of a variety of empirical Bayes methods to several well‐known model‐based prediction methods, including penalized regression, linear discriminant analysis, and Bayesian models with sparse or dense priors. We discuss “formal” empirical Bayes methods that maximize the marginal likelihood but also more informal approaches based on other data summaries. We contrast empirical Bayes to cross‐validation and full Bayes and discuss hybrid approaches. To study the relation between the quality of an empirical Bayes estimator and p, the number of variables, we consider a simple empirical Bayes estimator in a linear model setting. We argue that empirical Bayes is particularly useful when the prior contains multiple parameters, which model a priori information on variables termed “co‐data”. In particular, we present two novel examples that allow for co‐data: first, a Bayesian spike‐and‐slab setting that facilitates inclusion of multiple co‐data sources and types and, second, a hybrid empirical Bayes–full Bayes ridge regression approach for estimation of the posterior predictive interval.     

Robust Bayesian methodology with applications in credibility premium derivation and future claim size prediction

Robust Bayesian methodology deals with the problem of explaining uncertainty of the inputs (the prior, the model, and the loss function) and provides a breakthrough way to take into account the input’s variation. If the uncertainty is in terms of the prior knowledge, robust Bayesian analysis provides a way to consider the prior knowledge in terms of a class of priors \(\varGamma \) and derive some optimal rules. In this paper, we motivate utilizing robust Bayes methodology under the asymmetric general entropy loss function in insurance and pursue two main goals, namely (i) computing premiums and (ii) predicting a future claim size. To achieve the goals, we choose some classes of priors and deal with (i) Bayes and posterior regret gamma minimax premium computation, (ii) Bayes and posterior regret gamma minimax prediction of a future claim size under the general entropy loss. We also perform a prequential analysis and compare the performance of posterior regret gamma minimax predictors against the Bayes predictors.     

Asymptotic optimality of a two-stage procedure in Bayes sequential estimation

The problem of sequential estimation of the mean with quadratic loss and fixed cost per observation is considered within the Bayesian framework. Instead of fully sequential sampling, a two-stage sampling technique is introduced to solve the problem. The proposed two-stage procedure is robust in the sense that it does not depend on the distribution of outcome variables and the prior. It is shown to be asymptotically not worse than the optimal fixed-sample-size procedures for the arbitrary distributions, and to be asymptotically Bayes for the distributions of one-parameter exponential family.     

Constrained Bayes and Empirical Bayes Estimation with Balanced Loss Functions

This article develops constrained Bayes and empirical Bayes estimators under balanced loss functions. In the normal-normal example, estimators of the mean squared errors of the EB and constrained EB estimators are provided which are correct asymptotically up to O(m ^?1), m denoting the number of strata.     

Robustness of Bayes Prediction Under Error-in-Variables Superpopulation Model

In this article, we consider Bayes prediction in a finite population under the simple location error-in-variables superpopulation model. Bayes predictor of the finite population mean under Zellner's balanced loss function and the corresponding relative losses and relative savings loss are derived. The prior distribution of the unknown location parameter of the model is assumed to have a non-normal distribution belonging to the class of Edgeworth series distributions. Effects of non normality of the “true” prior distribution and that of a possible misspecification of the loss function on the Bayes predictor are illustrated for a hypothetical population.     

Empirical Bayes estimation for the mean of the selected normal population when there are additional data

ABSTRACT

This paper is concerned with the problem of estimation for the mean of the selected population from two normal populations with unknown means and common known variance in a Bayesian framework. The empirical Bayes estimator, when there are available additional observations, is derived and its bias and risk function are computed. The expected bias and risk of the empirical Bayes estimator and the intuitive estimator are compared. It is shown that the empirical Bayes estimator is asymptotically optimal and especially dominates the intuitive estimator in terms of Bayes risk, with respect to any normal prior. Also, the Bayesian correlation between the mean of the selected population (random parameter) and some interested estimators are obtained and compared.     

Nonparametric Bayes estimation of gap-time distribution with recurrent event data

Nonparametric Bayes (NPB) estimation of the gap-time survivor function governing the time to occurrence of a recurrent event in the presence of censoring is considered. In our Bayesian approach, the gap-time distribution, denoted by F, has a Dirichlet process prior with parameter α. We derive NPB and nonparametric empirical Bayes (NPEB) estimators of the survivor function F?=1?F and construct point-wise credible intervals. The resulting Bayes estimator of F? extends that based on single-event right-censored data, and the PL-type estimator is a limiting case of this Bayes estimator. Through simulation studies, we demonstrate that the PL-type estimator has smaller biases but higher root-mean-squared errors (RMSEs) than those of the NPB and the NPEB estimators. Even in the case of a mis-specified prior measure parameter α, the NPB and the NPEB estimators have smaller RMSEs than the PL-type estimator, indicating robustness of the NPB and NPEB estimators. In addition, the NPB and NPEB estimators are smoother (in some sense) than the PL-type estimator.     

Inference on proportional hazard rate model parameter under Type-I progressively hybrid censoring scheme

ABSTRACT

In this paper, under Type-I progressive hybrid censoring sample, we obtain maximum likelihood estimator of unknown parameter when the parent distribution belongs to proportional hazard rate family. We derive the conditional probability density function of the maximum likelihood estimator using moment-generating function technique. The exact confidence interval is obtained and compared by conducting a Monte Carlo simulation study for burr Type XII distribution. Finally, we obtain the Bayes and posterior regret gamma minimax estimates of the parameter under a precautionary loss function with precautionary index k = 2 and compare their behavior via a Monte Carlo simulation study.     

Inference based on progressively censored sample from Pareto population

Abstract

In this paper, we assume that the lifetimes have a two-parameter Pareto distribution and discuss some results of progressive Type-II censored sample. We obtain maximum likelihood estimators and Bayes estimators of the unknown parameters under squared error loss and a precautionary loss functions in progressively Type-II censored sample. Robust Bayes estimation of unknown parameters over three different classes of priors under progressively Type-II censored sample, squared error loss, and precautionary loss functions are obtained. We discuss estimation of unknown parameters on competing risks progressive Type-II censoring. Finally, we consider the problem of estimating the common scale parameter of two Pareto distributions when samples are progressively Type-II censored.     

Bayesian predictive distribution for a Poisson model with a parametric restriction

Abstract

Predictive probability estimation for a Poisson distribution is addressed when the parameter space is restricted. The Bayesian predictive probability against the prior on the restricted space is compared with the non-restricted Bayes predictive probability. It is shown that the former predictive probability dominates the latter under some conditions when the predictive probabilities are evaluated by the risk function relative to the Kullback-Leibler divergence. This result is proved by first showing the corresponding dominance result for estimating the restricted parameter and then translating it into the framework of predictive probability estimation.     

Admissible minimax estimators for the shape parameter of Topp–Leone distribution

Abstract

The shape parameter of Topp–Leone distribution is estimated in this article from the Bayesian viewpoint under the assumption of known scale parameter. Bayes and empirical Bayes estimates of the unknown parameter are proposed under non informative and suitable conjugate priors. These estimates are derived under the assumption of squared and linear-exponential error loss functions. The risk functions of the proposed estimates are derived in analytical forms. It is shown that the proposed estimates are minimax and admissible. The consistency of the proposed estimates under the squared error loss function is also proved. Numerical examples are provided.     

Bayes prediction of Poisson regression superpopulation mean with a non gamma prior

We consider Khamis' (1960) Laguerre expansion with gamma weight function as a class of “near-gamma” priors (K-prior) to obtain the Bayes predictor of a finite population mean under the Poisson regression superpopulation model using Zellner's balanced loss function (BLF). Kullback–Leibler (K-L) distance between gamma and some K-priors is tabulated to examine the quantitative prior robustness. Some numerical investigations are also conducted to illustrate the effects of a change in skewness and/or kurtosis on the Bayes predictor and the corresponding minimal Bayes predictive expected loss (MBPEL). Loss robustness with respect to the class of BLFs is also examined in terms of relative savings loss (RSL).     

A Generalized Bayes Rule for Prediction

In the case of prior knowledge about the unknown parameter, the Bayesian predictive density coincides with the Bayes estimator for the true density in the sense of the Kullback-Leibler divergence, but this is no longer true if we consider another loss function. In this paper we present a generalized Bayes rule to obtain Bayes density estimators with respect to any α-divergence, including the Kullback-Leibler divergence and the Hellinger distance. For curved exponential models, we study the asymptotic behaviour of these predictive densities. We show that, whatever prior we use, the generalized Bayes rule improves (in a non-Bayesian sense) the estimative density corresponding to a bias modification of the maximum likelihood estimator. It gives rise to a correspondence between choosing a prior density for the generalized Bayes rule and fixing a bias for the maximum likelihood estimator in the classical setting. A criterion for comparing and selecting prior densities is also given.     

Bayes estimation for exponential distributions with common location parameter and applications to multi-state reliability models

This paper considers the estimation of the stress–strength reliability of a multi-state component or of a multi-state system where its states depend on the ratio of the strength and stress variables through a kernel function. The article presents a Bayesian approach assuming the stress and strength as exponentially distributed with a common location parameter but different scale parameters. We show that the limits of the Bayes estimators of both location and scale parameters under suitable priors are the maximum likelihood estimators as given by Ghosh and Razmpour [15 M. Ghosh and A. Razmpour, Estimation of the common location parameter of several exponentials, Sankhyā, Ser. A 46 (1984), pp. 383–394. [Google Scholar]]. We use the Bayes estimators to determine the multi-state stress–strength reliability of a system having states between 0 and 1. We derive the uniformly minimum variance unbiased estimators of the reliability function. Interval estimation using the bootstrap method is also considered. Under the squared error loss function and linex loss function, risk comparison of the reliability estimators is carried out using extensive simulations.