On Progressively Type-II Censored Two-parameter Rayleigh Distribution

Recently, Rayleigh distribution has received considerable attention in the statistical literature. In this article, we consider the point and interval estimation of the functions of the unknown parameters of a two-parameter Rayleigh distribution. First, we obtain the maximum likelihood estimators (MLEs) of the unknown parameters. The MLEs cannot be obtained in explicit forms, and we propose to use the maximization of the profile log-likelihood function to compute the MLEs. We further consider the Bayesian inference of the unknown parameters. The Bayes’ estimates and the associated credible intervals cannot be obtained in closed forms. We use the importance sampling technique to approximate (compute) the Bayes’ estimates and the associated credible intervals. For comparison purposes, we have also used the exact method to compute the Bayes’ estimates and the corresponding credible intervals. Monte Carlo simulations are performed to compare the performances of the proposed method, and one dataset has been analyzed for illustrative purposes. We further consider the Bayes’ prediction problem based on the observed samples, and provide the appropriate predictive intervals. A data example has been provided for illustrative purposes.     

Analysis of hybrid censored competing risks data

In this paper, we consider the analysis of hybrid censored competing risks data, based on Cox's latent failure time model assumptions. It is assumed that lifetime distributions of latent causes of failure follow Weibull distribution with the same shape parameter, but different scale parameters. Maximum likelihood estimators (MLEs) of the unknown parameters can be obtained by solving a one-dimensional optimization problem, and we propose a fixed-point type algorithm to solve this optimization problem. Approximate MLEs have been proposed based on Taylor series expansion, and they have explicit expressions. Bayesian inference of the unknown parameters are obtained based on the assumption that the shape parameter has a log-concave prior density function, and for the given shape parameter, the scale parameters have Beta–Gamma priors. We propose to use Markov Chain Monte Carlo samples to compute Bayes estimates and also to construct highest posterior density credible intervals. Monte Carlo simulations are performed to investigate the performances of the different estimators, and two data sets have been analysed for illustrative purposes.     

On classical and Bayesian order restricted inference for multiple exponential step stress model

In this article, we consider the multiple step stress model based on the cumulative exposure model assumption. Here, it is assumed that for a given stress level, the lifetime of the experimental units follows exponential distribution and the expected lifetime decreases as the stress level increases. We mainly focus on the order restricted inference of the unknown parameters of the lifetime distributions. First we consider the order restricted maximum likelihood estimators (MLEs) of the model parameters. It is well known that the order restricted MLEs cannot be obtained in explicit forms. We propose an algorithm that stops in finite number of steps and it provides the MLEs. We further consider the Bayes estimates and the associated credible intervals under the squared error loss function. Due to the absence of explicit form of the Bayes estimates, we propose to use the importance sampling technique to compute Bayes estimates. We provide an extensive simulation study in case of three stress levels mainly to see the performance of the proposed methods. Finally the analysis of one real data set has been provided for illustrative purposes.     

Analysis of Type-II hybrid censored competing risks data

Kundu and Gupta [Analysis of hybrid life-tests in presence of competing risks. Metrica. 2007;65:159–170] provided the analysis of Type-I hybrid censored competing risks data, when the lifetime distributions of the competing cause of failures follows exponential distribution. In this paper, we consider the analysis of Type-II hybrid censored competing risks data. It is assumed that latent lifetime distributions of the competing causes of failures follow independent exponential distributions with different scale parameters. It is observed that the maximum likelihood estimators of the unknown parameters do not always exist. We propose the modified estimators of the scale parameters, which coincide with the corresponding maximum likelihood estimators when they exist, and asymptotically they are equivalent. We obtain the exact distribution of the proposed estimators. Using the exact distributions of the proposed estimators, associated confidence intervals are obtained. The asymptotic and bootstrap confidence intervals of the unknown parameters are also provided. Further, Bayesian inference of some unknown parametric functions under a very flexible Beta-Gamma prior is considered. Bayes estimators and associated credible intervals of the unknown parameters are obtained using the Monte Carlo method. Extensive Monte Carlo simulations are performed to see the effectiveness of the proposed estimators and one real data set has been analysed for the illustrative purposes. It is observed that the proposed model and the method work quite well for this data set.     

Bayes estimation and prediction of the two-parameter gamma distribution

In this article, the Bayes estimates of two-parameter gamma distribution are considered. It is well known that the Bayes estimators of the two-parameter gamma distribution do not have compact form. In this paper, it is assumed that the scale parameter has a gamma prior and the shape parameter has any log-concave prior, and they are independently distributed. Under the above priors, we use Gibbs sampling technique to generate samples from the posterior density function. Based on the generated samples, we can compute the Bayes estimates of the unknown parameters and can also construct HPD credible intervals. We also compute the approximate Bayes estimates using Lindley's approximation under the assumption of gamma priors of the shape parameter. Monte Carlo simulations are performed to compare the performances of the Bayes estimators with the classical estimators. One data analysis is performed for illustrative purposes. We further discuss the Bayesian prediction of future observation based on the observed sample and it is seen that the Gibbs sampling technique can be used quite effectively for estimating the posterior predictive density and also for constructing predictive intervals of the order statistics from the future sample.     

On the simple step-stress model for two-parameter exponential distribution

In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes.     

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.     

Parameter and reliability estimation for an exponentiated half-logistic distribution under progressive type II censoring

In this article, we deal with a two-parameter exponentiated half-logistic distribution. We consider the estimation of unknown parameters, the associated reliability function and the hazard rate function under progressive Type II censoring. Maximum likelihood estimates (M LEs) are proposed for unknown quantities. Bayes estimates are derived with respect to squared error, linex and entropy loss functions. Approximate explicit expressions for all Bayes estimates are obtained using the Lindley method. We also use importance sampling scheme to compute the Bayes estimates. Markov Chain Monte Carlo samples are further used to produce credible intervals for the unknown parameters. Asymptotic confidence intervals are constructed using the normality property of the MLEs. For comparison purposes, bootstrap-p and bootstrap-t confidence intervals are also constructed. A comprehensive numerical study is performed to compare the proposed estimates. Finally, a real-life data set is analysed to illustrate the proposed methods of estimation.     

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

Bayesian inference and prediction of order statistics for a Type-II censored Weibull distribution

This paper describes the Bayesian inference and prediction of the two-parameter Weibull distribution when the data are Type-II censored data. The aim of this paper is twofold. First we consider the Bayesian inference of the unknown parameters under different loss functions. The Bayes estimates cannot be obtained in closed form. We use Gibbs sampling procedure to draw Markov Chain Monte Carlo (MCMC) samples and it has been used to compute the Bayes estimates and also to construct symmetric credible intervals. Further we consider the Bayes prediction of the future order statistics based on the observed sample. We consider the posterior predictive density of the future observations and also construct a predictive interval with a given coverage probability. Monte Carlo simulations are performed to compare different methods and one data analysis is performed for illustration purposes.     

Analyzing the dynamic system model with discrete failure time distribution

The present study deals with the method of estimation of the parameters of k-components load-sharing parallel system model in which each component’s failure time distribution is assumed to be geometric. The maximum likelihood estimates of the load-share parameters with their standard errors are obtained. (1 − γ) 100% joint, Bonferroni simultaneous and two bootstrap confidence intervals for the parameters have been constructed. Further, recognizing the fact that life testing experiments are time consuming, it seems realistic to consider the load-share parameters to be random variable. Therefore, Bayes estimates along with their standard errors of the parameters are obtained by assuming Jeffrey’s invariant and gamma priors for the unknown parameters. Since, Bayes estimators can not be found in closed form expressions, Tierney and Kadane’s approximation method have been used to compute Bayes estimates and standard errors of the parameters. Markov Chain Monte Carlo technique such as Gibbs sampler is also used to obtain Bayes estimates and highest posterior density credible intervals of the load-share parameters. Metropolis–Hastings algorithm is used to generate samples from the posterior distributions of the unknown parameters.     

Inferences on Stress-Strength Reliability from Lindley Distributions

This article deals with the estimation of the stress-strength parameter R = P(Y < X) when X and Y are independent Lindley random variables with different shape parameters. The uniformly minimum variance unbiased estimator has explicit expression, however, its exact or asymptotic distribution is very difficult to obtain. The maximum likelihood estimator of the unknown parameter can also be obtained in explicit form. We obtain the asymptotic distribution of the maximum likelihood estimator and it can be used to construct confidence interval of R. Different parametric bootstrap confidence intervals are also proposed. Bayes estimator and the associated credible interval based on independent gamma priors on the unknown parameters are obtained using Monte Carlo methods. Different methods are compared using simulations and one data analysis has been performed for illustrative purposes.     

Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times

ABSTRACT

The maximum likelihood and Bayesian approaches for estimating the parameters and the prediction of future record values for the Kumaraswamy distribution has been considered when the lower record values along with the number of observations following the record values (inter-record-times) have been observed. The Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. The Bayes and the maximum likelihood estimates are compared by using the estimated risk through Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record values arising from the Kumaraswamy distribution based on record values with their corresponding inter-record times and only record values. The comparison of the derived predictors are carried out by using Monte Carlo simulations. Real data are analysed for an illustration of the findings.     

Statistical inference based on generalized Lindley record values

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.     

Bayesian Analysis of Different Hybrid and Progressive Life Tests

Type-I and Type-II censoring schemes are the widely used censoring schemes available for life testing experiments. A mixture of Type-I and Type-II censoring schemes is known as a hybrid censoring scheme. Different hybrid censoring schemes have been introduced in recent years. In the last few years, a progressive censoring scheme has also received considerable attention. In this article, we mainly consider the Bayesian inference of the unknown parameters of two-parameter exponential distribution under different hybrid and progressive censoring schemes. It is observed that in general the Bayes estimate and the associated credible interval of any function of the unknown parameters, cannot be obtained in explicit form. We propose to use the Monte Carlo sampling procedure to compute the Bayes estimate and also to construct the associated credible interval. Monte Carlo Simulation experiments have been performed to see the effectiveness of the proposed method in case of Type-I hybrid censored samples. The performances are quite satisfactory. One data analysis has been performed for illustrative purposes.     

Bayesian estimation based on progressive Type-II censoring from two-parameter bathtub-shaped lifetime model: an Markov chain Monte Carlo approach

In this paper, maximum likelihood and Bayes estimators of the parameters, reliability and hazard functions have been obtained for two-parameter bathtub-shaped lifetime distribution when sample is available from progressive Type-II censoring scheme. The Markov chain Monte Carlo (MCMC) method is used to compute the Bayes estimates of the model parameters. It has been assumed that the parameters have gamma priors and they are independently distributed. Gibbs within the Metropolis–Hasting algorithm has been applied to generate MCMC samples from the posterior density function. Based on the generated samples, the Bayes estimates and highest posterior density credible intervals of the unknown parameters as well as reliability and hazard functions have been computed. The results of Bayes estimators are obtained under both the balanced-squared error loss and balanced linear-exponential (BLINEX) loss. Moreover, based on the asymptotic normality of the maximum likelihood estimators the approximate confidence intervals (CIs) are obtained. In order to construct the asymptotic CI of the reliability and hazard functions, we need to find the variance of them, which are approximated by delta and Bootstrap methods. Two real data sets have been analyzed to demonstrate how the proposed methods can be used in practice.     

Statistical analysis of middle censored competing risks data with exponential distribution

In this paper, we consider some problems of estimation and reconstruction based on middle censored competing risks data. It is assumed that the lifetime distributions of the latent failure times are independent and exponential distributed with different parameters and also that the censoring mechanism is independent. The maximum likelihood estimators (MLEs) of the unknown parameters are obtained. We then use the asymptotic distribution of the MLEs to construct approximate confidence intervals. Based on gamma priors, Lindley's approximation method is applied to obtain the Bayesian estimates of the unknown parameters under squared error loss function. Since it is not possible to construct the credible intervals, we propose and implement the Gibbs sampling technique to construct the credible intervals. Several point reconstructors for failure time of censored units are provided. Finally, a simulation study is given by Monte-Carlo simulations to evaluate the performances of the different methods and a data set is analysed to illustrate the proposed procedures.     

On adaptive progressively Type-II censored competing risks data

In this article, a competing risks model based on exponential distributions is considered under the adaptive Type-II progressively censoring scheme introduced by Ng et al. [2009, Naval Research Logistics 56:687-698], for life testing or reliability experiment. Moreover, we assumed that some causes of failures are unknown. The maximum likelihood estimators (MLEs) of unknown parameters are established. The exact conditional and the asymptotic distributions of the obtained estimators are derived to construct the confidence intervals as well as the two different bootstraps of different unknown parameters. Under suitable priors on the unknown parameters, Bayes estimates and the corresponding two sides of Bayesian probability intervals are obtained. Also, for the purpose of evaluating the average bias and mean square error of the MLEs, and comparing the confidence intervals based on all mentioned methods, a simulation study was carried out. Finally, we present one real dataset to conduct the proposed methods.     

Estimation and prediction for Chen distribution with bathtub shape under progressive censoring

We consider estimation of the unknown parameters of Chen distribution [Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statist Probab Lett. 2000;49:155–161] with bathtub shape using progressive-censored samples. We obtain maximum likelihood estimates by making use of an expectation–maximization algorithm. Different Bayes estimates are derived under squared error and balanced squared error loss functions. It is observed that the associated posterior distribution appears in an intractable form. So we have used an approximation method to compute these estimates. A Metropolis–Hasting algorithm is also proposed and some more approximate Bayes estimates are obtained. Asymptotic confidence interval is constructed using observed Fisher information matrix. Bootstrap intervals are proposed as well. Sample generated from MH algorithm are further used in the construction of HPD intervals. Finally, we have obtained prediction intervals and estimates for future observations in one- and two-sample situations. A numerical study is conducted to compare the performance of proposed methods using simulations. Finally, we analyse real data sets for illustration purposes.     

Bayesian inference for the randomly censored Weibull distribution

In this paper, we consider the Bayesian inference of the unknown parameters of the randomly censored Weibull distribution. A joint conjugate prior on the model parameters does not exist; we assume that the parameters have independent gamma priors. Since closed-form expressions for the Bayes estimators cannot be obtained, we use Lindley's approximation, importance sampling and Gibbs sampling techniques to obtain the approximate Bayes estimates and the corresponding credible intervals. A simulation study is performed to observe the behaviour of the proposed estimators. A real data analysis is presented for illustrative purposes.