E-Bayesian estimation of reliability characteristics of two-parameter bathtub-shaped lifetime distribution with application

This article presents the expected Bayesian (E-Bayesian) estimation of the scale parameter, reliability and failure rate functions of two-parameter bathtub-shaped lifetime distribution under type-II censoring data with. Squared error loss function and gamma distribution as a conjugate prior distribution for the unknown parameter are used to obtain the E-Bayesian estimators. Also, three different prior distributions for the hyperparameters for the E-Bayesian estimators are considered. Some properties of the E-Bayesian estimators are studied. Using minimum mean square error criteria, a simulation study is conducted to compare the performance of the E-Bayesian estimators and the corresponding Bayes and maximum likelihood estimators. A real data set is analysed to show the applicability of the different proposed estimators. The numerical results show that the E-Bayesian estimators perform better than the classical and Bayesian estimators.     

Bayes‐conditional Control Charts for Weibull Quantiles under Type II Censoring

In this paper, we develop a Bayesian approach for monitoring Weibull quantiles under Type II censoring when prior information is negligible relative to the data. The posterior median of quantiles is considered as the monitored statistic. A method based on the relationship between Bayesian and conditional limits under an appropriate prior distribution is proposed to obtain the posterior median of quantiles in closed form. A pivotal quantity based on the monitored statistic is proposed, and its distribution is conditionally derived. Then, the Bayes‐conditional control limits are proposed. For the proposed charts, the probability of out‐of‐control can be derived without use of simulation. The performance of the Bayes‐conditional charts is compared with the bootstrap charts through the simulation methods. The results show that to monitor the first quantiles, the lower‐sided Bayes‐conditional charts perform better than bootstrap charts in detecting a downward shift caused by decreasing in the shape parameter. Finally, an illustrative example is provided. Copyright © 2016 John Wiley & Sons, Ltd.     

Inference for dependence competing risks with partially observed failure causes from bivariate Gompertz distribution under generalized progressive hybrid censoring

Competing risks model is considered with dependence causes of failure in this paper. When the latent failure times are distributed by a bivariate Gompertz model, statistical inference for the unknown model parameters is studied from classical and Bayesian approaches, respectively. Under a generalized progressive hybrid censoring, maximum likelihood estimators of the unknown parameters together with the associated existence and uniqueness are established, and the approximate confidence intervals are also obtained based on asymptotic likelihood theory via the observed Fisher information matrix. Moreover, Bayes estimates and the highest posterior density credible intervals of the unknown parameters are also provided based on a flexible Gamma–Dirichlet prior, and Monte Carlo sampling method is also derived to compute associated estimates. Finally, simulation studies and a real-life example are given for illustration purposes.     

A general Bayes weibull inference model for accelerated life testing

This article presents the development of a general Bayes inference model for accelerated life testing. The failure times at a constant stress level are assumed to belong to a Weibull distribution, but the specification of strict adherence to a parametric time-transformation function is not required. Rather, prior information is used to indirectly define a multivariate prior distribution for the scale parameters at the various stress levels and the common shape parameter. Using the approach, Bayes point estimates as well as probability statements for use-stress (and accelerated) life parameters may be inferred from a host of testing scenarios. The inference procedure accommodates both the interval data sampling strategy and type I censored sampling strategy for the collection of ALT test data. The inference procedure uses the well-known MCMC (Markov Chain Monte Carlo) methods to derive posterior approximations. The approach is illustrated with an example.     

Bayesian estimation of mixed Weibull distributions

Estimation of mixed Weibull distribution by maximum likelihood estimation and other methods is frequently difficult due to unstable estimates arising from limited data. Bayesian techniques can stabilize these estimates through the priors, but there is no closed-form conjugate family for the Weibull distribution. This paper reduces the number of numeric integrations required for using Bayesian estimation on mixed Weibull situations from five to two, thus making it a more feasible approach to the typical user. It also examines the robustness of the Bayesian estimates under a variety of different prior distributions. It is found that Bayesian estimation can improve accuracy over the MLE for situations with low mixture ratios so long as the prior on the weak subpopulation's characteristic life has an expected value less than or equal to the true characteristic life.     

A Bayesian life test sampling plan for products with Weibull lifetime distribution sold under warranty

A Bayesian life test sampling plan is considered for products with Weibull lifetime distribution which are sold under a warranty policy. It is assumed that the shape parameter of the distribution is a known constant, but the scale parameter is a random variable varying from lot to lot according to a known prior distribution. A cost model is constructed which involves three cost components; test cost, accept cost, and reject cost. A method of finding optimal sampling plans which minimize the expected average cost per lot is presented and sensitivity analyses for the parameters of the lifetime and prior distributions are performed.     

Non-parametric empirical Bayes procedure

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.     

Bayesian estimation of the 3-parameter inverse gaussian distribution

The three-parameter inverse Gaussian distribution is used as an alternative model for the three parameter lognormal, gamma and Weibull distributions for reliability problems. In this paper Bayes estimates of the parameters and reliability function of a three parameter inverse Gaussian distribution are obtained. Posterior variance estimates are compared with the variance of their maximum likelihood counterparts. Numerical examples are given.     

Statistical Inference From Censored Weihull Samples

In life testing experiments it is a fairly common practice to terminate the experiment before all items have failed. The Weibull distribution is often used as a model for the observations and when a computer is available maximum likelihood estimation of the parameters is to be recommended. The tables presented in this paper enable one to set confidence limits on the parameters and the reliability based on the maximum likelihood estimates for selected censoring and sample sizes.

It is also observed that, as in the case with no censoring, the maximum likelihood estimator of the reliability is very nearly unbiased and its variance is near the Cramér-Rao lower bound, Unbiasing factors for the maximum likelihood estimator of the shape parameter are given.     

Inference for step-stress plan with Khamis-Higgins model under type-II censored Weibull data

The accelerated life testing (ALT) is frequently used in examining the component reliability and acceptance testing. The ALT is carried out by exposing the unit to higher stress levels in order to observe data faster than those are producing under the normal conditions. The simple step-stress model based on type-II censoring Weibull lifetimes is studied here. In addition, the lifetimes satisfy Khamis-Higgins model assumption. In this paper, Bayesian approaches are developed for estimating the model parameters and predicting times to failure of future censored of the simple step-stress model from Weibull distribution using Khamis-Higgins model. The main goal of this work consists of two parts. First, the Bayesian estimation of the unknown parameters involved in the model is considered by adopting Devroye method to generate log-concave densities within sampling-based algorithm under different loss functions. The Bayes and highest posterior density credible intervals are then established. Second, the estimation of the posterior predictive density of the future lifetimes are discussed to obtain the point and prediction intervals with a given coverage probability. Monte Carlo simulation is performed to check the efficiency of the developed procedures and analyze a real data set for illustrative purposes.     

Reliability estimation of a multicomponent stress-strength model for unit Gompertz distribution under progressive Type II censoring

We consider the problem of estimating multicomponent stress-strength (MSS) reliability under progressive Type II censoring when stress and strength variables follow unit Gompertz distributions with common scale parameter. We estimate MSS reliability under frequentist and Bayesian approaches. Bayes estimates are obtained by using Lindley approximation and Metropolis-Hastings algorithm methods. Further, we obtain uniformly minimum variance unbiased estimates of the reliability when common scale parameter is known. Asymptotic, bootstrap confidence interval and highest posterior density credible intervals have been constructed. We perform Monte Carlo simulations to compare the performance of proposed estimates and also present a discussion. Finally, three real data sets are analyzed for illustrative purposes.     

A new variable control chart under failure‐censored reliability tests for Weibull distribution

In this paper, we propose a new variable control chart under type II or failure‐censored reliability tests by assuming that the lifetime of a part follows the Weibull distribution with fixed and stable shape parameter. The purpose is to monitor the mean and the variance of a Weibull process. In fact, the mean and the variance are related to the scale parameter. The necessary measures are given to calculate the average run length (ARL) for in‐control and shifted processes. The tables of ARLs are presented for various shift constants and specified parameters. A simulation study is given to show the performance of the proposed control chart. The efficiency of the proposed control chart is compared with a control chart based on the conditional expected value under type II censoring. An example is also given for the illustration purpose.     

Upper Bounds for the Power of Invariant Tests for the Exponential Distribution with Weibull Alternatives

The problem of testing for the exponential distribution (with scale, or both location and scale parameters unknown) against Weibull alternatives is considered. Upper bounds for the power of any invariant test are presented. Tests based upon the maximum likelihood estimator of the shape parameter, or a modification of it, are given which virtually achieve these bounds.     

Statistical inference on progressive-stress accelerated life testing for the logistic exponential distribution under progressive type-II censoring

The accelerated life testing (ALT) is an efficient approach and has been used in several fields to obtain failure time data of test units in a much shorter time than testing at normal operating conditions. In this article, a progressive-stress ALT under progressive type-II censoring is considered when the lifetime of test units follows logistic exponential distribution. We assume that the scale parameter of the distribution satisfying the inverse power law. First, the maximum likelihood estimates of the model parameters and their approximate confidence intervals are obtained. Next, we obtain Bayes estimators under squared error loss function with the help of Metropolis-Hasting (MH) algorithm. We also derive highest posterior density (HPD) credible intervals of the model parameters. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation. Finally, one data set has been analyzed for illustrative purposes.     

Robust Regression using Probability Plots for Estimating the Weibull Shape Parameter

The Weibull shape parameter is important in reliability estimation as it characterizes the ageing property of the system. Hence, this parameter has to be estimated accurately. This paper presents a study of the efficiency of using robust regression methods over the ordinary least‐squares regression method based on a Weibull probability plot. The emphasis is on the estimation of the shape parameter of the two‐parameter Weibull distribution. Both the case of small data sets with outliers and the case of data sets with multiple‐censoring are considered. Maximum‐likelihood estimation is also compared with linear regression methods. Simulation results show that robust regression is an effective method in reducing bias and it performs well in most cases. Copyright © 2006 John Wiley & Sons, Ltd.     

The Bayes procedure in exponential reliability family models using conjugate convex tent prior family

A Bayes procedure for estimating the unknown parameter indexed to some of one parameter exponential family distributions is presented. In such a procedure, we shall use a new conjugate prior family that we shall call a conjugate convex tent family. A member of this family could be constructed by assuming a little information about the unknown parameter. Some of the needed parameters for constructing prior density function are the values r, p and q. Bayes estimators for using a priori symmetrical convex tent density can be obtained as special cases of the present work. Numerical simulation study is introduced by using the Monte Carlo method. In this study we have investigated the influence of the prior parameters r, p and q on the accuracy of the Bayes estimators.     

An Exact Asymptotically Efficient Confidence Bound for Reliability in the Case of the Weibull Distribution

This paper presents a simple method for obtaining exact lower confidence bounds for reliabilities (tail probabilities) for items whose life times follow a Weibull distribution where both the “shape” and “scale” parameters are unknown. These confidence bounds are obtained both for the censored and non-censored cases and are asymptotically efficient. They are exact even for small sample sizes in that they attain the desired confidence level precisely. The case of an additional unknown “location” or “shift” parameter is also discussed in the large sample case. Tables are given of exact and asymptotic lower confidence bounds for the reliability for sample sizes of 10, 15, 20, 30, 50 and 100 for various censoring fractions.     

An Addendum to “Linear Estimation of the Weibull Parameters”

Linear, asymptotically normal and efficient estimators are given for the shape parameter of the two parameter Weibull distribution when the scale parameter is known and for the log of the scale parameter when the shape parameter is known. The weights of the ordered observations and other constants needed for these estimators are readily obtainable from a previous article of the author.     

Reliability analysis of the dynamic system for the Chen model through sequential order statistics

Recently, the two-parameter Chen distribution has widely been used for reliability studies in various engineering fields. In this article, we have developed various statistical inferences on the composite dynamic system, assuming Chen distribution as a baseline model. In this dynamic system, failure of a component induces a higher load on the surviving components and thus increases component hazard rate through a power-trend process. The classical and Bayesian point estimates of the unknown parameters of the composite system are obtained by the method of maximum likelihood and Markov chain Monte Carlo techniques, respectively. In the Bayesian framework, we have used gamma priors to obtain Bayes estimates of unknown parameters under the squared error and generalized entropy loss functions. The interval estimates of the baseline reliability function are obtained by using the Fisher information matrix and Bayesian method. A parametric hypothesis test is presented to test whether the failed components change the hazard rate function. A compact simulation study is carried out to examine the behavior of the proposed estimation methods. Finally, one real data analysis is performed for illustrative purposes.     

Joint estimation of E–N curves and their scatter using evolutionary algorithms

We present an approach for estimating E–N curves and their scatter. The scatter of a number of load cycles to failure at an arbitrary amplitude-strain level is modelled using a two-parametric Weibull distribution with the constant shape parameter β and the scale parameter η dependent on the strain amplitude by the Coffin–Manson equation. In this way the E–N curve and its scatter can be described using five parameters: the four parameters of the Coffin–Manson equation for the scale parameter of the Weibull distribution and the shape parameter of the Weibull distribution. The objective was to estimate these five parameters, which are generally unknown (since the data from the literature are manly known only for the median E–N curves), on the basis of the known fatigue-life data to obtain not only the trend of the E–N curve, but also its scatter. In order to estimate these parameters on the basis of the fatigue-life data, two evolutionary algorithms were applied: a real-valued genetic algorithm (GA) and the differential ant-stigmergy algorithm (DASA). In the article a mathematical background of the approach is presented and applied to 27 test cases of simulated fatigue-life data and one real case of experimentally obtained fatigue-life data. The results are analysed and discussed.