Simultaneous selection of extreme populations:a bayesian approach

The problem of simultaneously selecting two non-empty subsets, S_Land S_U, of k populations which contain the lower extreme population (LEP) and the upper extreme population (UEP), respectively, is considered. Unknown parameters θ₁,…,θ_kcharacterize the populations π₁,…,π_kand the populations associated with θ_[1]=min θ_i. and θ_[k]= max θ_i. are called the LEP and the UEP, respectively. It is assumed that the underlying distributions possess the monotone likelihood ratio property and that the prior distribution of θ= (θ₁,…,θ_k) is exchangeable. The Bayes rule with respect to a general loss function is obtained. Bayes rule with respect to a semi-additive and non-negative loss function is also determined and it is shown that it is minimax and admissible. When the selected subsets are required to be disjoint, it shown that the Bayes rule with respect to a specific loss function can be obtained by comparing certain computable integrals, Application to normal distributions with unknown means θ₁,…,θ_kand a common known variance is also considered.     

Bayes and robust Bayes predictions in a subfamily of scale parameters under a precautionary loss function

ABSTRACT

This paper deals with Bayes, robust Bayes, and minimax predictions in a subfamily of scale parameters under an asymmetric precautionary loss function. In Bayesian statistical inference, the goal is to obtain optimal rules under a specified loss function and an explicit prior distribution over the parameter space. However, in practice, we are not able to specify the prior totally or when a problem must be solved by two statisticians, they may agree on the choice of the prior but not the values of the hyperparameters. A common approach to the prior uncertainty in Bayesian analysis is to choose a class of prior distributions and compute some functional quantity. This is known as Robust Bayesian analysis which provides a way to consider the prior knowledge in terms of a class of priors Γ for global prevention against bad choices of hyperparameters. Under a scale invariant precautionary loss function, we deal with robust Bayes predictions of Y based on X. We carried out a simulation study and a real data analysis to illustrate the practical utility of the prediction procedure.     

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 rules for selecting good populations

A problem of selecting populations better than a control is considered. When the populations are uniformly distributed, empirical Bayes rules are derived for a linear loss function for both the known control parameter and the unknown control parameter cases. When the priors are assumed to have bounded supports, empirical Bayes rules for selecting good populations are derived for distributions with truncation parameters (i.e. the form of the pdf is f(x|θ)= p_i(x)c_i(θ)I_{(0, θ)}(x)). Monte Carlo studies are carried out which determine the minimum sample sizes needed to make the relative errors less than ε for given ε-values.     

Alternative Bayes factors for model selection

Several alternative Bayes factors have been recently proposed in order to solve the problem of the extreme sensitivity of the Bayes factor to the priors of models under comparison. Specifically, the impossibility of using the Bayes factor with standard noninformative priors for model comparison has led to the introduction of new automatic criteria, such as the posterior Bayes factor (Aitkin 1991), the intrinsic Bayes factors (Berger and Pericchi 1996b) and the fractional Bayes factor (O'Hagan 1995). We derive some interesting properties of the fractional Bayes factor that provide justifications for its use additional to the ones given by O'Hagan. We further argue that the use of the fractional Bayes factor, originally introduced to cope with improper priors, is also useful in a robust analysis. Finally, using usual classes of priors, we compare several alternative Bayes factors for the problem of testing the point null hypothesis in the univariate normal model.     

Expansion estimation by Bayes rules

In the problem of estimating a location parameter in any symmetric unimodal location parameter model, we demonstrate that Bayes rules with respect to squared error loss can be expanders for some priors that belong to the family of all symmetric priors. That generalizes the results obtained by DasGupta and Rubin for the one dimensional case. We also consider symmetric priors which either have an appropriate point mass at 0 or are unimodal, and prove that under the latter condition all Bayes rules are shrinkers. Results of such nature are important, for example, in wavelet based function estimation and data denoising, where shrinkage of wavelet coefficients is associated with smoothing the data. We illustrate the results using FIAT stock market data.     

On selecting the best treatment in a generalized linear model

The problem of selecting the best treatment is studied under generalized linear models. For certain balanced designs, it is shown that simple rules are Bayes with respect to any non-informative prior on the treatment effects under any monotone invariant loss. When the nuisance parameters such as block effects are assumed to follow a uniform (improper) prior or a normal prior, Bayes rules are obtained for the normal linear model under more suitable balanced designs, keeping the generality of the loss and the generality of the non-informativeness on the prior of the treatment effects. These results are extended to certain types of informative priors on the treatment effects. When the designs are unbalanced, algorithms based on the Gibbs sampler and the Laplace method are provided to compute the Bayes rules.     

Lower bounds on bayes factors for a linear regression model

This paper develops a Bayesian analysis of testing linear parameter restictions in a normal regression model. Instead of searching for a suitable "objective" prior for such a test, several interesting classes of informative priors are postulated, and the lower bounds on the Bayes factors are derived.     

The equivalence of Bayes and robust Bayes estimators for various loss functions

The problem of Bayes and robust Bayes estimation for various bounded and/or symmetric loss functions in a normal model with conjugate and non-informative prior distributions is considered. The prior distribution is not fully specified and covers the conjugate family of priors. It is of interest to know that the Bayes and robust Bayes estimators for symmetric losses are the same as those for the standard square-error loss function.     

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).     

ESTIMATION OF EXPONENTIATED WEIBULL SHAPE PARAMETERS UNDER LINEX LOSS FUNCTION

ABSTRACT

The paper deals with Bayes estimation of the exponentiated Weibull shape parameters under linex loss function when independent non-informative type of priors are available for the parameters. Generalized maximum likelihood estimators have also been obtained. Performances of the proposed Bayes estimator, generalized maximum likelihood estimators, posterior mean (i.e., Bayes estimator under squared error loss function) and maximum likelihood estimators have been studied on the basis of their risks under linex loss function. The comparison is based on a simulation study because the expressions for risk functions of these estimators cannot be obtained in nice closed forms.     

A note on james-stein and bayes empiricl bayes estimators

In an empirical Bayes decision problem, a simple class of estimators is constructed that dominate the James-Stein

estimator, A prior distribution A is placed on a restricted (normal) class G of priors to produce a Bayes empirical Bayes estimator, The Bayes empirical Bayes estimator is smooth, admissible, and asymptotically optimal. For certain A rate of convergence to minimum Bayes risk is 0(n^-1)uniformly on G. The results of a Monte Carlo study are presented to demonstrate the favorable risk bebhavior of the Bayes estimator In comparison with other competitors including the James-Stein estimator.     

Simultaneous selection and estimation in general linear models

The problem of selecting the largest treatment parameter, and simultaneously estimating the selected treatment parameter, in a general linear model is considered in the decision theoretic Bayes approach. Both cases, where the error variance is known or unknown, are included. Bayes decision rules are derived for noninformative priors and for normal priors. The problem of finding Bayes designs, i.e. designs that have minimum Bayes risk, within a given class of designs is also discussed.     

Bayes admissible estimation of the means in Poisson decomposable graphical models

We investigate the Bayes estimation of the means in Poisson decomposable graphical models. Some classes of Bayes estimators are provided which improve on the maximum likelihood estimator under the normalized squared error loss. Both proper and improper priors are included in the proposed classes of priors. Concerning the generalized Bayes estimators with respect to the improper priors, we address their admissibility.     

Intrinsic Priors for Testing Two Normal Means with Intrinsic Bayes Factors

ABSTRACT

In this article we consider the problem of comparing two normal means with unknown common variance using a Bayesian approach. Conventional Bayes factors with improper non informative priors are not well defined. The intrinsic Bayes factors are used to overcome such a difficulty. We derive intrinsic priors whose Bayes factors are asymptotically equivalent to the corresponding intrinsic Bayes factors. We illustrate our results with numerical examples.     

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.     

INTRINSIC PRIORS FOR TWO-SAMPLE TESTS IN NORMAL POPULATIONS

ABSTRACT

There have been considerable amounts of work regarding the development of various default Bayes factors in model selection and hypothesis testing. Two commonly used criteria, the intrinsic Bayes factor and the fractional Bayes factor are compared to test two independent normal means and variances. We also derive several intrinsic priors whose Bayes factors are asymptotically equivalent to the respective Bayes factors. We demonstrate our results in simulated datasets.     

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.     

Subset Selection Toward Optimizing the Best Performance at a Second Stage

In the search for the best of n candidates, two-stage procedures of the following type are in common use. In a first stage, weak candidates are removed, and the subset of promising candidates is then further examined. At a second stage, the best of the candidates in the subset is selected. In this article, optimization is not aimed at the parameter with largest value but rather at the best performance of the selected candidates at Stage 2. Under a normal model, a new procedure based on posterior percentiles is derived using a Bayes approach, where nonsymmetric normal (proper and improper) priors are applied. Comparisons are made with two other procedures frequently used in selection decisions. The three procedures and their performances are illustrated with data from a recent recruitment process at a Midwestern university.     

Empirical Bayes estimators in hierarchical models with mixture priors

We consider subgroup analyses within the framework of hierarchical modeling and empirical Bayes (EB) methodology for general priors, thereby generalizing the normal–normal model. By doing this one obtains greater flexibility in modeling. We focus on mixture priors, that is, on the situation where group effects are exchangeable within clusters of subgroups only. We establish theoretical results on accuracy, precision, shrinkage and selection bias of EB estimators under the general priors. The impact of model misspecification is investigated and the applicability of the methodology is illustrated with datasets from the (medical) literature.