Bayesian estimation and prediction intervals for a Rayleigh distribution under a conjugate prior

This paper is an effort to obtain Bayes estimators of Rayleigh parameter and its associated risk based on a conjugate prior (square root inverted gamma prior) with respect to both symmetric loss function (squared error loss), and asymmetric loss function (precautionary loss function). We also derive the highest posterior density (HPD) interval for the Rayleigh parameter as well as the HPD prediction intervals for a future observation from this distribution. An illustrative example to test how the Rayleigh distribution fits a real data set is presented. Finally, Monte Carlo simulations are performed to compare the performances of the Bayes estimates under different conditions.     

Bayesian analysis for the two-parameter Pareto distribution based on record values and times

Doostparast and Balakrishnan (Pareto record-based analysis, Statistics, under review) recently developed optimal confidence intervals as well as uniformly most powerful tests for one- and two-sided hypotheses concerning shape and scale parameters, for the two-parameter Pareto distribution based on record data. In this paper, on the basis of record values and inter-record times from the two-parameter Pareto distribution, maximum-likelihood and Bayes estimators as well as credible regions are developed for the two parameters of the Pareto distribution. For illustrative purposes, a data set on annual wages of a sample of production-line workers in a large industrial firm is analysed using the proposed procedures.     

Bayesian analysis of threshold autoregressions

A nonasymptotic Bayesian approach is developed for analysis of data from threshold autoregressive processes with two regimes. Using the conditional likelihood function, the marginal posterior distribution for each of the parameters is derived along with posterior means and variances. A test for linear functions of the autoregressive coefficients is presented. The approach presented uses a posterior p-value averaged over the values of the threshold. The one-step ahead predictive distribution is derived along with the predictive mean and variance. In addition, equivalent results are derived conditional upon a value of the threshold. A numerical example is presented to illustrate the approach.     

An objective Bayesian analysis of the two-parameter half-logistic distribution based on progressively type-II censored samples

This paper develops an objective Bayesian analysis method for estimating unknown parameters of the half-logistic distribution when a sample is available from the progressively Type-II censoring scheme. Noninformative priors such as Jeffreys and reference priors are derived. In addition, derived priors are checked to determine whether they satisfy probability-matching criteria. The Metropolis–Hasting algorithm is applied to generate Markov chain Monte Carlo samples from these posterior density functions because marginal posterior density functions of each parameter cannot be expressed in an explicit form. Monte Carlo simulations are conducted to investigate frequentist properties of estimated models under noninformative priors. For illustration purposes, a real data set is presented, and the quality of models under noninformative priors is evaluated through posterior predictive checking.     

On priors that match posterior and frequentist distribution functions

In the multiparameter case, this paper characterizes priors so as to match, up to o(n^-1/2), the posterior joint cumulative distribution function (c.d.f.) of a posterior standardized version of the parametric vector with the corresponding frequentist c.d.f.     

Bayesian prediction for two-parameter weibull lifetime models

Prediction methods for the two-parameter Weibull distribution are computationally complicated. This paper shows how an approximate Bayesian method can be used to simplify the computation and presents a simpler alternative for computing prediction bounds derived classically by Lawless (1973).     

Bayesian prediction of order statistics based on k-record values from exponential distribution

Bayesian prediction of order statistics as well as the mean of a future sample based on observed record values from an exponential distribution are discussed. Several Bayesian prediction intervals and point predictors are derived. Finally, some numerical computations are presented for illustrating all the proposed inferential procedures.     

Bayesian inference for the randomly censored Burr-type XII distribution

The article presents the Bayesian inference for the parameters of randomly censored Burr-type XII distribution with proportional hazards. The joint conjugate prior of the proposed model parameters does not exist; we consider two different systems of priors for Bayesian estimation. The explicit forms of the Bayes estimators are not possible; we use Lindley's method to obtain the Bayes estimates. However, it is not possible to obtain the Bayesian credible intervals with Lindley's method; we suggest the Gibbs sampling procedure for this purpose. Numerical experiments are performed to check the properties of the different estimators. The proposed methodology is applied to a real-life data for illustrative purposes. The Bayes estimators are compared with the Maximum likelihood estimators via numerical experiments and real data analysis. The model is validated using posterior predictive simulation in order to ascertain its appropriateness.     

A default Bayesian procedure for the generalized Pareto distribution

The generalized Pareto distribution is used to model the exceedances over a threshold in a number of fields, including the analysis of environmental extreme events and financial data analysis. We use this model in a default Bayesian framework where no prior information is available on unknown model parameters. Using a large simulation study, we compare the performance of our posterior estimations of parameters with other methods proposed in the literature. We show that our procedure also allows to make inferences in other quantities of interest in extreme value analysis without asymptotic arguments. We apply the proposed methodology to a real data set.     

On estimation of R=P(Y<X) for exponential distribution under progressive type-II censoring

This paper deals with the estimation of the stress–strength parameter R=P(Y<X), when X and Y are independent exponential random variables, and the data obtained from both distributions are progressively type-II censored. The uniformly minimum variance unbiased estimator and the maximum-likelihood estimator (MLE) are obtained for the stress–strength parameter. Based on the exact distribution of the MLE of R, an exact confidence interval of R has been obtained. Bayes estimate of R and the associated credible interval are also obtained under the assumption of independent inverse gamma priors. An extensive computer simulation is used to compare the performances of the proposed estimators. One data analysis has been performed for illustrative purpose.     

Predictive analysis for joint progressive censoring plans: a Bayesian approach

Comparative lifetime experiments are of particular importance in production processes when one wishes to determine the relative merits of several competing products with regard to their reliability. This paper confines itself to the data obtained by running a joint progressive Type-II censoring plan on samples in a combined manner. The problem of Bayesian predicting failure times of surviving units is discussed in details when parent populations are exponential. Two real data sets are analyzed in order to illustrate all the inferential procedures developed here. When destructive experiments under a censoring scheme finished, the researchers are usually interested to estimate remaining lifetimes of surviving units for sequel experiments. Findings of this paper are useful for these purposes specially when samples are non-homogeneous such as those taken from industrial storages.     

Bayes estimation for the Block and Basu bivariate and multivariate Weibull distributions

Block and Basu bivariate exponential distribution is one of the most popular absolute continuous bivariate distributions. Recently, Kundu and Gupta [A class of absolute continuous bivariate distributions. Statist Methodol. 2010;7:464–477] introduced Block and Basu bivariate Weibull (BBBW) distribution, which is a generalization of the Block and Basu bivariate exponential distribution, and provided the maximum likelihood estimators using EM algorithm. In this paper, we consider the Bayesian inference of the unknown parameters of the BBBW distribution. The Bayes estimators are obtained with respect to the squared error loss function, and the prior distributions allow for prior dependence among the unknown parameters. Prior independence also can be obtained as a special case. It is observed that the Bayes estimators of the unknown parameters cannot be obtained in explicit forms. We propose to use the importance sampling technique to compute the Bayes estimates and also to construct the associated highest posterior density credible intervals. The analysis of two data sets has been performed for illustrative purposes. The performances of the proposed estimators are quite satisfactory. Finally, we generalize the results for the multivariate case.     

Small sample set estimation of a baseline ranking

Ranking objects by a panel of judges is commonly used in situations where objective attributes cannot easily be measured or interpreted. Under the assumption that the judge independently arrive at their rankings by making pair wise comparisons among the objects in an attempt to reproduce a common baseline ranking w₀, we develop and explore confidence regions and Bayesian highest posterior density credible regions for w₀ with emphasis on very small sample sizes.     

Bayesian and frequentist prediction limits for the Poisson distribution

Prediction limits for Poisson distribution are useful in real life when predicting the occurrences of some phenomena, for example, the number of infections from a disease per year among school children, or the number of hospitalizations per year among patients with cardiovascular disease. In order to allocate the right resources and to estimate the associated cost, one would want to know the worst (i.e., an upper limit) and the best (i.e., the lower limit) scenarios. Under the Poisson distribution, we construct the optimal frequentist and Bayesian prediction limits, and assess frequentist properties of the Bayesian prediction limits. We show that Bayesian upper prediction limit derived from uniform prior distribution and Bayesian lower prediction limit derived from modified Jeffreys non informative prior coincide with their respective frequentist limits. This is not the case for the Bayesian lower prediction limit derived from a uniform prior and the Bayesian upper prediction limit derived from a modified Jeffreys prior distribution. Furthermore, it is shown that not all Bayesian prediction limits derived from a proper prior can be interpreted in a frequentist context. Using a counterexample, we state a sufficient condition and show that Bayesian prediction limits derived from proper priors satisfying our condition cannot be interpreted in a frequentist context. Analysis of simulated data and data on Atlantic tropical storm occurrences are presented.     

Approximate bayes estimation of reliability of two parameter inverse gaussian distribution

This paper extends the result of Padgett (1981) and gives a Bayes estimate of the reliability function of two-parameter inverse Gaussian distribution using Jeffreys' non-informative joint prior and a squared error loss fun ction . A numerical example is given. Based on a Monte Carlo simulation, Bayes estimator of reliability is compared with its maximum likelihood counterpart.     

Bayesian analysis of the generalized gamma distribution using non-informative priors

The Generalized gamma (GG) distribution plays an important role in statistical analysis. For this distribution, we derive non-informative priors using formal rules, such as Jeffreys prior, maximal data information prior and reference priors. We have shown that these most popular formal rules with natural ordering of parameters, lead to priors with improper posteriors. This problem is overcome by considering a prior averaging approach discussed in Berger et al. [Overall objective priors. Bayesian Analysis. 2015;10(1):189–221]. The obtained hybrid Jeffreys-reference prior is invariant under one-to-one transformations and yields a proper posterior distribution. We obtained good frequentist properties of the proposed prior using a detailed simulation study. Finally, an analysis of the maximum annual discharge of the river Rhine at Lobith is presented.     

A Hybrid Approximation Bayesian Test of Variance Components for Longitudinal Data

The test of variance components of possibly correlated random effects in generalized linear mixed models (GLMMs) can be used to examine if there exists heterogeneous effects. The Bayesian test with Bayes factors offers a flexible method. In this article, we focus on the performance of Bayesian tests under three reference priors and a conjugate prior: an approximate uniform shrinkage prior, modified approximate Jeffreys' prior, half-normal unit information prior and Wishart prior. To compute Bayes factors, we propose a hybrid approximation approach combining a simulated version of Laplace's method and importance sampling techniques to test the variance components in GLMMs.     

Bayesian inference on P(X > Y) in bivariate Rayleigh model

In the literature, assuming independence of random variables X and Y, statistical estimation of the stress–strength parameter R = P(X > Y) is intensively investigated. However, in some real applications, the strength variable X could be highly dependent on the stress variable Y. In this paper, unlike the common practice in the literature, we discuss on estimation of the parameter R where more realistically X and Y are dependent random variables distributed as bivariate Rayleigh model. We derive the Bayes estimates and highest posterior density credible intervals of the parameters using suitable priors on the parameters. Because there are not closed forms for the Bayes estimates, we will use an approximation based on Laplace method and a Markov Chain Monte Carlo technique to obtain the Bayes estimate of R and unknown parameters. Finally, simulation studies are conducted in order to evaluate the performances of the proposed estimators and analysis of two data sets are provided.