A Bivariate Geometric Distribution with Applications to Reliability

This article studies a bivariate geometric distribution (BGD) as a plausible reliability model. Maximum likelihood and Bayes estimators of parameters and various reliability characteristics are obtained. Approximations to the mean, variance, and Bayes risk of these estimators have been derived using Taylor's expansion. A Monte-Carlo simulation study has been performed to compare these estimators. At the end, the theory is illustrated with a real data set example of accidents.     

Bayes estimators of the reliability of the logistic distribution

Bayes estimators of the reliability function of the logistic distribution are obtained using the methods of Lindley (1980) and Tierney & Kadane (1986). Squared-error and log-odds squared-error loss functions are used. A numerical example is presented. Comparisons are made between these two procedures, based on a Monte Carlo simulation study.     

Estimating the exponential reliability function for a system in series with type II censored data

The problem of estimating the one parameter exponential reliability function for a system composed of l componentes in series is considered. Under the type II censoring scheme, the Bayes nature of the minimum variance unbiased estimator is demonstrated and the admissibility of related generalized Bayes estimators is established. For the one component case, the best unbiased estimator is admissible.     

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

In this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.     

Bayes estimation of reliability for the two-parameter lognormal distribution

Bayes estimators of reliability for the lognormal failure distribution with two parameters (M,∑) are obtained both for informative priors of normal-gamma type and for the vague prior of Jeffreys. The estimators are in terms of the t-distribution function. The Bayes estimators are compared with the maximum likelihood and minimum variance unbiased estimators of reliabil-ity using Monte Carlo simulations.     

Reliability estimation in a multicomponent stress–strength model under generalized half-normal distribution based on progressive type-II censoring

In this paper, based on progressively Type-II censored samples, the problem of estimation of multicomponent stress–strength reliability under generalized half-normal (GHN) distribution is considered. The reliability of a k-component stress-strength system is estimated when both stress and strength variates are assumed to have a GHN distribution with various cases of same and different shape and scale parameters. Different methods such as the maximum likelihood estimates (MLEs) and Bayes estimation are discussed. The expectation maximization algorithm and approximate maximum likelihood methods are proposed to compute the MLE of reliability. The Lindley's approximation method, as well as Metropolis–Hastings algorithm, are applied to compute Bayes estimates. The performance of the proposed procedures is also demonstrated via a Monte Carlo simulation study and an illustrative example.     

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.     

The umvue and bayes estimate of reliability of mixed failure time distribution

We study the reliability estimates of the non-standard mixture of degenerate (degenerated at zero) and exponential distributions. The Uniformly Minimum Variance Unbiased Estimator (UMVUE) and Bayes estimator of the reliability for some selective prior when the mixing proportion is known and unknown are derived. The Bayes risk is computed for each Bayes estimator of the reliability. A simulated study is carried out to assess the performance of the estimators alongwith the true and Maximum Likelihood Estimate (MLE) of the reliability. An example from Vannman (1991) is also discussed at the end of the paper.     

Comparison of linex and quadratic bayes estimators foe the rayleigh distribution

In this paper, the Bayes estimators for the parameter, the reliability function, and failure rate function of the Rayleigh distribution are obtained when based on complete or type II censored samples. Some types of the linex loss function are used. Comparieons in terms of risks of those under linex loss and squared error loss function with Bayes estimators relative to squared error loss function are made, Numerical example and simulation example are included.     

Point estimation of the stress–strength reliability parameter for parallel system with independent and non-identical components

In this article, the estimation problem of the multicomponent stress–strength reliability parameter is considered where the stress and the strength systems have arbitrary fixed numbers of independent and non-identical parallel components. It is assumed that the distribution functions of the stress and the strength components satisfy the proportional reversed hazard rate model. The study is done in more details when the baseline distributions are exponential. Maximum likelihood and uniformly minimum variance unbiased estimators are obtained and compared. Also, Bayes and empirical Bayes estimators are discussed and Monte Carlo simulations are carried out to compare their performances.     

Estimation of the scale parameter of the Rayleigh distribution with multiply type–II censored sample

Based on a multiply type-II censored sample, the maximum likelihood estimator (MLE) and Bayes estimator for the scale parameter and the reliability function of the Rayleigh distribution are derived. However, since the MLE does not exist an explicit form, an approximate MLE which is the maximizer of an approximate likelihood function will be given. The comparisons among estimators are investigated through Monte Carlo simulations. An illustrative example with the real data concerning the 23 ball bearing in the life test is presented.     

Classical and Bayesian estimation of stress-strength reliability in type II censored Pareto distributions

ABSTRACT

In this paper, the stress-strength reliability, R, is estimated in type II censored samples from Pareto distributions. The classical inference includes obtaining the maximum likelihood estimator, an exact confidence interval, and the confidence intervals based on Wald and signed log-likelihood ratio statistics. Bayesian inference includes obtaining Bayes estimator, equi-tailed credible interval, and highest posterior density (HPD) interval given both informative and non-informative prior distributions. Bayes estimator of R is obtained using four methods: Lindley's approximation, Tierney-Kadane method, Monte Carlo integration, and MCMC. Also, we compare the proposed methods by simulation study and provide a real example to illustrate them.     

On the Bayesian analysis of the mixture of power function distribution using the complete and the censored sample

The power function distribution is often used to study the electrical component reliability. In this paper, we model a heterogeneous population using the two-component mixture of the power function distribution. A comprehensive simulation scheme including a large number of parameter points is followed to highlight the properties and behavior of the estimates in terms of sample size, censoring rate, parameters size and the proportion of the components of the mixture. The parameters of the power function mixture are estimated and compared using the Bayes estimates. A simulated mixture data with censored observations is generated by probabilistic mixing for the computational purposes. Elegant closed form expressions for the Bayes estimators and their variances are derived for the censored sample as well as for the complete sample. Some interesting comparison and properties of the estimates are observed and presented. The system of three non-linear equations, required to be solved iteratively for the computations of maximum likelihood (ML) estimates, is derived. The complete sample expressions for the ML estimates and for their variances are also given. The components of the information matrix are constructed as well. Uninformative as well as informative priors are assumed for the derivation of the Bayes estimators. A real-life mixture data example has also been discussed. The posterior predictive distribution with the informative Gamma prior is derived, and the equations required to find the lower and upper limits of the predictive intervals are constructed. The Bayes estimates are evaluated under the squared error loss function.     

Use of approximate bayesian methods for the block and basu bivariate exponential distribution

Summary In this paper, we present a Bayesian analysis of the bivariate exponential distribution of Block and Basu (1974) assuming different prior densities for the parameters of the model and considering Laplace's method to obtain approximate marginal posterior and posterior moments of interest. We also find approximate Bayes estimators for the reliability of two-component systems at a specified timet ₀ considering series and parallel systems. We illustrate the proposed methodology with a generated data set.     

Inference and prediction for pareto progressively censored data

Progressively censored data from a classical Pareto distribution are to be used to make inferences about its shape and precision parameters and the reliability function. An approximation form due to Tierney and Kadane (1986) is used for obtaining the Bayes estimates. Bayesian prediction of further observations from this distribution is also considered. When the Bayesian approach is concerned, conjugate priors for either the one or the two parameters cases are considered. To illustrate the given procedures, a numerical example and a simulation study are given.     

Bayesian analysis of head and neck cancer data using generalized inverse Lindley stress–strength reliability model

In this article, we present the analysis of head and neck cancer data using generalized inverse Lindley stress–strength reliability model. We propose Bayes estimators for estimating P(X > Y), when X and Y represent survival times of two groups of cancer patients observed under different therapies. The X and Y are assumed to be independent generalized inverse Lindley random variables with common shape parameter. Bayes estimators are obtained under the considerations of symmetric and asymmetric loss functions assuming independent gamma priors. Since posterior becomes complex and does not possess closed form expressions for Bayes estimators, Lindley’s approximation and Markov Chain Monte Carlo techniques are utilized for Bayesian computation. An extensive simulation experiment is carried out to compare the performances of Bayes estimators with the maximum likelihood estimators on the basis of simulated risks. Asymptotic, bootstrap, and Bayesian credible intervals are also computed for the P(X > Y).     

On designing an acceptance sampling plan for the Pareto lifetime model

In this paper, the design of reliability sampling plans for the Pareto lifetime model under progressive Type-II right censoring is considered. Sampling plans are derived using the decision theoretic approach with a suitable loss or cost function that consists of sampling cost, rejection cost, and acceptance cost. The decision rule is based on the estimated reliability function. Plans are constructed within the Bayesian context using the natural conjugate prior. Simulations for evaluating the Bayes risk are carried out and the optimal sampling plans are reported for various sample sizes, observed number of failures and removal probabilities.     

Estimation of parameters of inverse Weibull distribution and application to multi-component stress-strength model

The problem of estimation of the parameters of two-parameter inverse Weibull distributions has been considered. We establish existence and uniqueness of the maximum likelihood estimators of the scale and shape parameters. We derive Bayes estimators of the parameters under the entropy loss function. Hierarchical Bayes estimator, equivariant estimator and a class of minimax estimators are derived when shape parameter is known. Ordered Bayes estimators using information about second population are also derived. We investigate the reliability of multi-component stress-strength model using classical and Bayesian approaches. Risk comparison of the classical and Bayes estimators is done using Monte Carlo simulations. Applications of the proposed estimators are shown using real data sets.     

Estimation from Burr type XII distribution using progressive first-failure censored data

In this paper, a new life test plan called a progressively first-failure-censoring scheme introduced by Wu and Ku? [On estimation based on progressive first-failure-censored sampling, Comput. Statist. Data Anal. 53(10) (2009), pp. 3659–3670] is considered. Based on this type of censoring, the maximum likelihood (ML) and Bayes estimates for some survival time parameters namely reliability and hazard functions, as well as the parameters of the Burr-XII distribution are obtained. The Bayes estimators relative to both the symmetric and asymmetric loss functions are discussed. We use the conjugate prior for the one-shape parameter and discrete prior for the other parameter. Exact and approximate confidence intervals with the exact confidence region for the two-shape parameters are derived. A numerical example using the real data set is provided to illustrate the proposed estimation methods developed here. The ML and the different Bayes estimates are compared via a Monte Carlo simulation study.     

On finite mixture of two-component gompertz lifetime model

The Gompertz distribution has been used as a growth model, especially in epidemiological and biomedical studies. Based on Type I and II censored samples from a heterogeneous population that can be represented by a finite mixture of two-component Gompertz lifetime model, the maximum likelihood and Bayes estimates of the parameters, reliability and hazard rate functions are obtained. An approximation form due to Lindley (1980) is used in obtaining the corresponding Bayes estimates. The maximum likelihood and Bayes estimates are comparedvia a Monte Carlo simulation study.