The Bayes rule of the parameter in (0,1) under Zhang’s loss function with an application to the beta-binomial model

Abstract

For the restricted parameter space (0,1), we propose Zhang’s loss function which satisfies all the 7 properties for a good loss function on (0,1). We then calculate the Bayes rule (estimator), the posterior expectation, the integrated risk, and the Bayes risk of the parameter in (0,1) under Zhang’s loss function. We also calculate the usual Bayes estimator under the squared error loss function, and the Bayes estimator has been proved to underestimate the Bayes estimator under Zhang’s loss function. Finally, the numerical simulations and a real data example of some monthly magazine exposure data exemplify our theoretical studies of two size relationships about the Bayes estimators and the Posterior Expected Zhang’s Losses (PEZLs).     

Some Applications of the Rao Distance to Shrinkage Estimators

Bayesian statistics is concerned with how prior information influence inferences. This article studies this problem by comparing the value of the Rao distance between prior and posterior normal distributions. Particular cases include the linear Bayes estimator, the mixed estimator, and ridge-type estimators.     

Bayesian estimation of sensitivity level and population proportion of a sensitive characteristic in a binary optional unrelated question RRT model

Sihm et al. (2016 Sihm, J. S., A. Chhabra, and S. N. Gupta. 2016. An optional unrelated question RRT model. Involve: A Journal of Mathematics 9 (2):195–209.[Crossref] , [Google Scholar]) proposed an unrelated question binary optional randomized response technique (RRT) model for estimating the proportion of population that possess a sensitive characteristic and the sensitivity level of the question. In our work, decision theoretic approach has been followed to obtain Bayes estimates of the two parameters along with their corresponding minimal Bayes posterior expected losses (BPEL) using beta prior and squared error loss function (SELF). Relative losses are also examined to compare the performances of the Bayes estimates with those of the classical estimates obtained by Sihm et al. (2016 Sihm, J. S., A. Chhabra, and S. N. Gupta. 2016. An optional unrelated question RRT model. Involve: A Journal of Mathematics 9 (2):195–209.[Crossref] , [Google Scholar]). The results obtained are illustrated with the help of real survey data using non informative prior.     

Une condition nécessaire d'admissibilité et ses conséquences sur les estimateurs à rétrécisseur de la moyenne d'un vecteur normal

Considering exponential families of distributions, we estimate parameters which are not the natural parameters. We prove that the admissible estimators of these parameters are limits of Bayes estimators and can be expressed through a given functional form. An important particular case of this model pertains to the estimation of the mean of a multidimensional normal distribution when the variance is known up to a multiplicative factor. We deduce from the main result a necessry condition for the admissibility of matricial shrinkage estimators.     

Bayes Prediction for a Heteroscedastic Regression Superpopulation Model Using Balanced Loss Function

We consider Prais–Houthakker heteroscedastic normal regression model having variance of the dependent variable same as square of its expectation. Bayes predictors for the regression coefficient and the mean of a finite population are derived using Zellner's balanced loss function. Bayes predictive expected losses are obtained and compared with those of classical predictors and Bayes predictors under squared error loss function to examine their loss robustness.     

Bayesian estimation of parameters of inverse Weibull distribution

The present paper describes the Bayes estimators of parameters of inverse Weibull distribution for complete, type I and type II censored samples under general entropy and squared error loss functions. The proposed estimators have been compared on the basis of their simulated risks (average loss over sample space). A real-life data set is used to illustrate the results.     

A Note on the Estimation of Binomial Probabilities

I am concerned with the admissibility under quadratic loss of certain estimators of binomial probabilities. The minimum variance unbiased estimator is shown to be admissible for Pr(X = 0) and Pr(X = n), but it is inadmissible for Pr(X = k), where 0 < k < n. An example is given of an admissible maximum likelihood estimator (MLE). It is conjectured that the MLE is always admissible.     

Bayesian confidence estimation: an alternative approach

From a Bayesian point of wiew, the estimation of an unknown parameter can be interpreted (in many situations) as the problem of fixing a partition of the parameter space (by means of small intervals I₁,…I_k and choosing an interval I_j, from this partition, provided that sufficient information has been obtained. This idea is developed in a decision theory setting. If the Kolmogorov-Smirnov loss is used, it is proved that I_j is the best interval estimation if and only if its posterior probability is greater than or equal to 1/2     

Comparing Normal Random Samples,with Uncertainty About the Priors and Utilities

Abstract. We study the problem of deciding which of two normal random samples, at least one of them of small size, has greater expected value. Unlike in the standard Bayesian approach, in which a single prior distribution and a single loss function are declared, we assume that a set of plausible priors and a set of plausible loss functions are elicited from the expert (the client or the sponsor of the analysis). The choice of the sample that has greater expected value is based on equilibrium priors, allowing for an impasse if for some plausible priors and loss functions choosing one and for others the other sample is associated with smaller expected loss.     

Estimation of multiple gamma scale-parameters: bayes estimation subject to uniform domination

Simultaneous estimation of p gamma scale-parameters is considered under squared-error loss. The problem of minimizing, subject to uniform risk domination, the Bayes risk (or more generally the posterior expected loss) against certain conjugate or mixtures of conjugate priors is considered. Rather surprisingly, it is shown that the minimization can be done conditionally, thus avoiding variational arguments. Relative savings loss (and a posterior version thereof) are found, and it is found that in the most favorable situations, Bayesian robustness can be achieved without sacrificing substantial subjective Bayesian gains.     

Choosing Priors for Constrained Analysis of Variance: Methods Based on Training Data

Abstract. This article combines the best of both objective and subjective Bayesian inference in specifying priors for inequality and equality constrained analysis of variance models. Objectivity can be found in the use of training data to specify a prior distribution, subjectivity can be found in restrictions on the prior to formulate models. The aim of this article is to find the best model in a set of models specified using inequality and equality constraints on the model parameters. For the evaluation of the models an encompassing prior approach is used. The advantage of this approach is that only a prior for the unconstrained encompassing model needs to be specified. The priors for all constrained models can be derived from this encompassing prior. Different choices for this encompassing prior will be considered and evaluated.     

Bayesian Estimation of Regression Coefficients Under Extended Balanced Loss Function

Appreciating the desirability of simultaneously using both the criteria of goodness of fitted model and clustering of estimates around true parameter values, an extended version of the balanced loss function is presented and the Bayesian estimation of regression coefficients is discussed. The thus obtained optimal estimator is then compared with the least squares estimator and posterior mean vector with respect to the criteria like posterior expected loss, Bayes risk, bias vector, mean squared error matrix and risk function.     

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.     

PARAMETER ESTIMATION AND APPLICATIONS FOR A GENERALISATION OF THE BETA-BINOMIAL DISTRIBUTION

A three-parameter generalisation of the beta-binomial distribution (BBD) derived by Chandon (1976) is examined. We obtain the maximum likelihood estimates of the parameters and give the elements of the information matrix. To exhibit the applicability of the generalised distribution we show how it gives an improved fit over the BBD for magazine exposure and consumer purchasing data. Finally we derive an empirical Bayes estimate of a binomial proportion based on the generalised beta distribution used in this study.     

Some remarks on classical and bayesian reliability estimation of binomial and poisson distributions

The problems of estimating the reliability function and Pr{X₁+...+X_k ≤ Y} are considered. The random variables X’s and Y are assumed to follow binomial and Poisson distributions. Classical estimators available in the literature are discussed and Bayes estimators are derived. In order to obtain the estimators of these parametric functions, the basic role is played by the estimators of factorial moments of the two distributions.