Bayesian analysis of life-testing problems involving the weibull distribution

Credible and highest posterior density intervals for the reliability function and parameters of a two-parameter Weibull process are obtained and the estimates compared with their corresponding classical counterparts.     

On the estimation of the true probability of a correct selection

For ranking and selection problems, the true probabiIity of a correct selection P(CS) is unknown even if a selection is made under the indifference-zone approach. Thus to estimate the true P(CS) some Bayes estimators and a bootstrap estimator are proposed for two normcal populations with common known variance. Also a bootstrap estimator and a bootstrap confidence interval are proposed for normal populations with common unknown variance. Some comparisons between proposed estimators and some other known estimators are made via Monte Carlo simulations.     

Simultaneous Credible Bands for Latent Gaussian Models

Abstract. Deterministic Bayesian inference for latent Gaussian models has recently become available using integrated nested Laplace approximations (INLA). Applying the INLA‐methodology, marginal estimates for elements of the latent field can be computed efficiently, providing relevant summary statistics like posterior means, variances and pointwise credible intervals. In this article, we extend the use of INLA to joint inference and present an algorithm to derive analytical simultaneous credible bands for subsets of the latent field. The algorithm is based on approximating the joint distribution of the subsets by multivariate Gaussian mixtures. Additionally, we present a saddlepoint approximation to compute Bayesian contour probabilities, representing the posterior support of fixed parameter vectors of interest. We perform a simulation study and apply the given methods to two real examples.     

Bayesian estimation and prediction based on lognormal record values

In this paper we consider the problems of estimation and prediction when observed data from a lognormal distribution are based on lower record values and lower record values with inter-record times. We compute maximum likelihood estimates and asymptotic confidence intervals for model parameters. We also obtain Bayes estimates and the highest posterior density (HPD) intervals using noninformative and informative priors under square error and LINEX loss functions. Furthermore, for the problem of Bayesian prediction under one-sample and two-sample framework, we obtain predictive estimates and the associated predictive equal-tail and HPD intervals. Finally for illustration purpose a real data set is analyzed and simulation study is conducted to compare the methods of estimation and prediction.     

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.     

Small sample performance of jackknife confidence intervals for the james-stein estimator

The primary goal of this paper is to examine the small sample coverage probability and size of jackknife confidence intervals centered at a Stein-rule estimator. A Monte Carlo experiment is used to explore the coverage probabilities and lengths of nominal 90% and 95% delete-one and infinitesimal jackknife confidence intervals centered at the Stein-rule estimator; these are compared to those obtained using a bootstrap procedure.     

Exact inference on multiple exponential populations under a joint type-II progressive censoring scheme

ABSTRACT

Recently Mondal and Kundu [Mondal S, Kundu D. A new two sample type-II progressive censoring scheme. Commun Stat Theory Methods. 2018. doi:10.1080/03610926.2018.1472781] introduced a Type-II progressive censoring scheme for two populations. In this article, we extend the above scheme for more than two populations. The aim of this paper is to study the statistical inference under the multi-sample Type-II progressive censoring scheme, when the underlying distributions are exponential. We derive the maximum likelihood estimators (MLEs) of the unknown parameters when they exist and find out their exact distributions. The stochastic monotonicity of the MLEs has been established and this property can be used to construct exact confidence intervals of the parameters via pivoting the cumulative distribution functions of the MLEs. The distributional properties of the ordered failure times are also obtained. The Bayesian analysis of the unknown model parameters has been provided. The performances of the different methods have been examined by extensive Monte Carlo simulations. We analyse two data sets for illustrative purposes.     

Nonparametric estimation of the entropy using a ranked set sample

This article is concerned with nonparametric estimation of the entropy in ranked set sampling. Theoretical properties of the proposed estimator are studied. The proposed estimator is compared with the rival estimator in simple random sampling. The applications of the proposed estimator to the mutual information estimation as well as estimation of the Kullback–Leibler divergence are provided. Several Monté-Carlo simulation studies are conducted to examine the performance of the estimator. The results are applied to the longleaf pine (Pinus palustris) trees and the body fat percentage datasets to illustrate applicability of theoretical results.     

On the choice of the prior distribution in hypergeometric sampling

Information in a statistical procedure arising from sources other than sampling is called prior information, and its incorporation into the procedure forms the basis of the Bayesian approach to statistics. Under hypergeometric sampling, methodology is developed which quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information embedded in any prior distribution. The most conservative prior distribution from a specified class (each member of which carries the available prior information) is that prior distribution within the class over which the likelihood function has the greatest average domination. Four different families of prior distributions are developed by considering a Bayesian approach to the formation of lots. The most conservative prior distribution from each of the four families of prior distributions is determined and compared for the situation when no prior information is available. The results of the comparison advocate the use of the Polya (beta-binomial) prior distribution in hypergeometric sampling.     

Robust and efficient estimation of GARCH models based on Hellinger distance

It is well known that financial data frequently contain outlying observations. Almost all methods and techniques used to estimate GARCH models are likelihood-based and thus generally non-robust against outliers. Minimum distance method, as an important tool for statistical inferences and a competitive alternative for achieving robustness, has surprisingly not been well explored for GARCH models. In this paper, we proposed a minimum Hellinger distance estimator (MHDE) and a minimum profile Hellinger distance estimator (MPHDE), depending on whether the innovation distribution is specified or not, for estimating the parameters in GARCH models. The construction and investigation of the two estimators are quite involved due to the non-i.i.d. nature of data. We proved that the MHDE is a consistent estimator and derived its bias in explicit expression. For both of the proposed estimators, we demonstrated their finite-sample performance through simulation studies and compared with the well-established methods including MLE, Gaussian Quasi-MLE, Non-Gaussian Quasi-MLE and Least Absolute Deviation estimator. Our numerical results showed that MHDE and MPHDE have much better performance than MLE-based methods when data are contaminated while simultaneously they are very competitive when data is clean, which testified to the robustness and efficiency of the two proposed MHD-type estimations.     

Likelihood and bayesian approaches to inference for the stationary point of a quadratic response surface

In response surface analysis, a second order polynomial model is often used for inference on the stationary point of the response function. The standard confidence regions for the stationary point are due to Box & Hunter (1954). The authors propose an alternative parametrization, in which the stationary point is the parameter of interest; likelihood techniques and Bayesian analysis are then easier to perform. The authors also suggest an approximate method to get highest posterior density regions for the maximum point (not simply for the stationary point). Furthermore, they study the coverage probabilities of these Bayesian regions through simulations.     

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.     

Robust bayesian analysis given a lower bound on the probability of a set

Suppose that just the lower bound of the probability of a measurable subset K in the parameter space Ω is a priori known, when inferences are to be made about measurable subsets A in Ω. Instead of eliciting a unique prior distribution, consider the class Г of all the distributions compatible with such bound. Under mild regularity conditions about the likelihood function, the range of the posterior probability of any A is found, as the prior distribution varies in Г. Such ranges are analysed according to the robust Bayesian viewpoint. Furthermore, some characterising properties of the extended likelihood sets are proved. The prior distributions in Г are then considered as a neighbour class of an elicited prior, comparing likelihood sets and HPD in terms of robustness.     

Bayesian tests of a parameter shift under heteroscedasticity:weighted-t vs. double-t approaches

A highest posterior density interval for examining shifts in individual regression coefficients between subsamples with different error variances is derived from a weighted-t posterior density. Compared to a double-t approach, the weighted-t approach is computationally more efficient and allows a direct comparison to a sampling theory approach. In a numerical example the weighted-t, double-t and confidence interval approaches are compared.     

Small sample behavior of a robust heteroskedasticity consistent covariance matrix estimator

In heteroskedastic regression models, the least squares (OLS) covariance matrix estimator is inconsistent and inference is not reliable. To deal with inconsistency one can estimate the regression coefficients by OLS, and then implement a heteroskedasticity consistent covariance matrix (HCCM) estimator. Unfortunately the HCCM estimator is biased. The bias is reduced by implementing a robust regression, and by using the robust residuals to compute the HCCM estimator (RHCCM). A Monte-Carlo study analyzes the behavior of RHCCM and of other HCCM estimators, in the presence of systematic and random heteroskedasticity, and of outliers in the explanatory variables.     

Hierarchical bayes estimation of normal variances with application to a random effect model

The problem of simultaneously estimating p normal variances is investigated when the parameters are believed a priori to be similar in size. A hierarchical Bayes approach is employed and the resulting estimator is compared to common estimators used including one proposed by Box and Tiao (1973) using a Bayesian approach with a noninformative prior. The technique is then applied to estimate components of variance in the one way layout random effect model of the analysis of variance.     

Interval estimation of a small proportion via inverse sampling

On the basis of a negative binomial sampling scheme, we consider a uniformly most accurate upper confidence limit for a small but unknown proportion, such as the proportion of defectives in a manufacturing process. The optimal stopping rule, with reference to the twin criteria of the expected length of the confidence interval and the expected sample size, is investigated. The proposed confidence interval has also been compared with several others that have received attention in the recent literature.