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• Corrected an error (posterior risk of Bayesian estimation under the entropy loss function) in Ali et al. [10]. • Derived the corrected posterior risk of Bayesian estimation under the entropy loss function. • Derived posterior risk of Bayesian estimation for binomial distribution under the entropy and squared error loss function. • Monte Carlo simulation example and application example are provided for illustrative purposes. Scholars have discussed the Bayesian estimation and their corresponding posterior risks for the Lindley distribution parameter, respectively, using seven different loss functions, where the posterior risk under entropy loss function (ELF) is not correct. In this paper, we will derive the Bayesian estimation and its corrected posterior risk under the entropy loss function. Moreover, the Bayesian estimations and their posterior risks of binomial distribution parameter under the entropy loss function and squared error loss function are respectively developed. Monte Carlo simulation example and application example are provided for illustrative purposes, and results are compared based on posterior risk. [ABSTRACT FROM AUTHOR]

This study discusses the Bayesian and maximum likelihood estimation methods for analyzing the data from a 3-component mixture of Burr Type-XII probability distributions. The maximum likelihood estimators with their variances cannot be obtained in an explicit form and thus an iterative procedure is used to calculate them numerically. Contrary to this, elegant closed form algebraic expressions of Bayes estimators and their posterior risks are derived. Using the informative and noninformative priors, the posterior predictive distributions along with predictive intervals are also discussed. A method of eliciting hyperparameters using prior predictive distribution is also a part of this study. Some interesting properties (including, posterior risks and Bayesian predictive intervals) of Bayes estimators and their behavior across different sample sizes, left and right test termination times, informative prior versus Jeffreys noninformative prior, are provided via a detailed Monte Carlo simulation study. To assess the suitability and application of the proposed model, a real life data example is also discussed in this article. Based on the simulated results and real data application, it is concluded that the IP paired with DLF (SELF) is a more suitable for estimating mixing component. [ABSTRACT FROM AUTHOR]

As compared to simple models, the mixture models of underlying lifetime distributions are intuitively more appropriate and appealing to model the heterogeneous nature of process. This study focuses on the problem of estimating the parameters of a newly developed 3-component mixture of Burr Type-XII distributions using Type-I right censored data. Firstly, considering a Bayesian structure, some mathematical properties of a 3-component mixture of Burr Type-XII distributions are discussed. These mathematical properties include Bayes estimators and posterior risks for the unknown component and proportion parameters using the non-informative and the informative priors under squared error loss function, precautionary loss function and DeGroot loss function. Secondly, in case when no or little prior information is available, elicitation of hyperparameters is given. Also, the posterior predictive distribution for a future observation and the Bayesian predictive interval are constructed. Moreover, the limiting expressions for the Bayes estimators and posterior risks are derived. In addition, the performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. Finally, simulated datasets are designed for the different comparisons and the model is illustrated using the real data. [ABSTRACT FROM AUTHOR]

An engineering process is an output from a set of combined processes which may be homogeneous or heterogeneous. To study the lifetime of such processes, we need a model which can accommodate the nature of such processes. Single probability models are not capable of capturing the heterogeneity of nature. However, mixture models of some suitable lifetime distributions, have the potential to highlight such interesting feature. Due to time and cost constraint, in the most lifetime testing experiments, censoring is an unavoidable feature of most lifetime data sets. This article deals with the modeling of the heterogeneity existing in the lifetime processes using the mixture of the inverse Rayleigh distribution, and the spotlight is the Bayesian inference of the mixture model using non-informative (the Jeffreys and the uniform) and informative (gamma) priors. We are considering this particular distribution due to two reasons; the first one is due to its skewed behavior, i.e. in engineering processes, an engineer suspects that high failure rate in the beginning, but after continuous inspection, the failure goes down and the second reason is due to its vast application in many applied fields. A Gibbs sampling algorithm based on adaptive rejection sampling is designed for the posterior computation. A detailed simulation study is carried out to investigate the performance of the estimators based on different prior distributions. The posterior risks are evaluated under the squared error, the weighted, the quadratic, the entropy, the modified squared error and the precautionary loss functions. Posterior risks of the Bayes estimates are compared to explore the effect of prior information and loss functions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, mathematical properties of Lindley distribution via Bayesian approach are derived under different loss functions. These properties include: Bayes Estimators, posterior risks and failure rate function for simulation scheme. Elicitation of hyperparameters is also discussed. A real life application to waiting time data at the bank is also described for the developed procedures (also using WinBUGS). Results are compared on the basis of posterior risk. [Copyright &y& Elsevier]

In this paper we introduce an empirical Bayes procedure for estimating an unknown parameter, say θ. This procedure gives the empirical Bayes estimator for θ and its associated minimum posterior risk in closed forms without estimating the unknown prior density function of θ. In such procedure the posterior probability density function of θ is not required. A sufficient statistic for θ with conditional probability density function in the one parameter exponential family is required. Instead of estimating the unknown prior density function, the marginal density function of the sufficient statistic must be estimated. As special cases the empirical Bayes estimators and their respective minimum posterior risks of the failure rate for the exponential distribution, the unknown scale parameters of Weibull and gamma distributions are obtained in simple forms as special cases. Numerical results and a simulation study are introduced to (i) investigate how the number of available past experiments and the sample size of each influence the accuracy of the empirical Bayes estimator, (ii) make a comparison between the presented procedure and the Bayes procedure when the prior probability density function of the parameter θ is gamma. [Copyright &y& Elsevier]

For many Bayesians, the posterior risk (posterior expected loss), which is a function of the data and the prior information, is the only measure of accuracy of interest. It is shown here that the value of this measure of accuracy may increase (uncertainty increases) as more data are obtained. The value of the posterior variance, which the posterior risk when the loss function is the squared error one, may exceed the value of the initial measure of accuracy, which is the prior variance. A specific example is analyzed numerically using Fortran language. [Copyright &y& Elsevier]