Alaa H. Abdel-Hamid

Estimation in step-stress accelerated life tests for the exponentiated exponential distribution with type-I censoring

The step-stress accelerated life tests allow the experimenter to increase the stress levels at fixed times during the experiment. The lifetime of a product at any level of stress is assumed to have an exponentiated distribution, whose baseline distribution is a general class of distributions which includes, among others, Weibull, compound Weibull, Pareto, Gompertz, normal and logistic distributions. The scale parameter of the baseline distribution is assumed to be a log-linear function of the stress and a cumulative exposure model holds. Special attention is paid to an exponentiated exponential distribution. Based on type-I censoring, the maximum likelihood estimates of the parameters under consideration are obtained. A Monte Carlo simulation study is carried out to investigate the precision of the maximum likelihood estimates and to obtain the coverage probabilities of the bootstrap confidence intervals for the parameters involved. Finally, an example is presented to illustrate the two discussed methods of bootstrap confidence intervals.

Constant-partially accelerated life tests for Burr type-XII distribution with progressive type-II censoring

Based on progressively type-II censored samples, constant-partially accelerated life tests (PALTs) when the lifetime of items under use condition follow the two-parameter Burr type-XII (Burr(c,k)) distribution are considered. The likelihood equations of the involved parameters are derived and then reduced to a single nonlinear equation to be solved numerically to obtain the maximum likelihood estimates (MLEs) of the parameters. The observed Fisher information matrix, as well as the asymptotic variance-covariance matrix of the MLEs are derived. Approximate confidence intervals (CIs) for the parameters, based on normal approximation to the asymptotic distribution of MLEs, studentized-t and percentile bootstrap CIs are derived. A Monte Carlo simulation study is carried out to investigate the precision of the MLEs and to compare the performance of the CIs considered. Finally, two examples presented to illustrate our results are followed by conclusions.

Inference for a progressive stress model from Weibull distribution under progressive type-II censoring

Abstract Based on progressively type-II censored samples, this paper considers progressive stress accelerated life tests when the lifetime of an item under use condition follows the Weibull distribution with a scale parameter satisfying the inverse power law. It is assumed that the progressive stress is directly proportional to time and the cumulative exposure model for the effect of changing stress holds. Point estimation of the model parameters is obtained graphically by using Weibull probability paper plot that serves as a tool for model identification and also by using the maximum likelihood method. Interval estimation is performed by finding approximate confidence intervals (CIs) for the parameters as well as the studentized-t and percentile bootstrap CIs. Monte Carlo simulation study is carried out to investigate the precision of the estimates and compare the performance of CIs obtained. Finally, two examples are presented to illustrate our results.

Bayesian prediction for type-II progressive-censored data from the Rayleigh distribution under progressive-stress model

The two-sample prediction is considered to derive Bayesian prediction bounds for future order statistics based on progressively type-II censored informative samples obtained from progressive-stress accelerated life testing models. The lifetime of an item under the use condition stress is assumed to follow the Rayleigh distribution with a scale parameter satisfying the inverse power law. The informative and future samples are assumed to be obtained from the same population. Explicit forms for prediction bounds of the first future order statistic are obtained in the case of one unknown parameter. When two parameters are unknown, a simulation study is performed and numerical computations are carried out, based on three different progressive-censoring schemes. The coverage probabilities and average interval lengths of the confidence intervals are computed via a Monte Carlo simulation.

Step partially accelerated life tests under finite mixture models

In this paper, step partially accelerated life tests are considered when the lifetime of an item under use condition follows a finite mixture of distributions. The analysis is performed when each of the components follows a general class of distributions, which includes, among others, the Weibull, compound Weibull (or three-parameter Burr type XII), power function, Gompertz and compound Gompertz distributions. Based on type-I censoring, the maximum likelihood estimates (MLEs) of the mixing proportions, scale parameters and acceleration factor are obtained. Special attention is paid to a mixture of two exponential components. Simulation results are obtained to study the precision of MLEs.

Inference on progressive-stress model for the exponentiated exponential distribution under type-II progressive hybrid censoring

In this paper, progressive-stress accelerated life tests are applied when the lifetime of a product under design stress follows the exponentiated distribution [G(x)]α. The baseline distribution, G(x), follows a general class of distributions which includes, among others, Weibull, compound Weibull, power function, Pareto, Gompertz, compound Gompertz, normal and logistic distributions. The scale parameter of G(x) satisfies the inverse power law and the cumulative exposure model holds for the effect of changing stress. A special case for an exponentiated exponential distribution has been discussed. Using type-II progressive hybrid censoring and MCMC algorithm, Bayes estimates of the unknown parameters based on symmetric and asymmetric loss functions are obtained and compared with the maximum likelihood estimates. Normal approximation and bootstrap confidence intervals for the unknown parameters are obtained and compared via a simulation study.

Inference and Optimal Design Based on Step—Partially Accelerated Life Tests for the Generalized Pareto Distribution under Progressive Type-I Censoring

Various types of failure, censored and accelerated life tests, are commonly employed for life testing in some manufacturing industries and products that are highly reliable. In this article, we consider the tampered failure rate model as one of such types that relate the distribution under use condition to the distribution under accelerated condition. It is assumed that the lifetimes of products under use condition have generalized Pareto distribution as a lifetime model. Some estimation methods such as graphical, moments, probability weighted moments, and maximum likelihood estimation methods for the parameters are discussed based on progressively type-I censored data. The determination of optimal stress change time is discussed under two different criteria of optimality. Finally, a Monte Carlo simulation study is carried out to examine the performance of the estimation methods and the optimality criteria.

A new lifetime distribution for a series-parallel system: properties, applications and estimations under progressive type-II censoring

ABSTRACT A compound class of zero truncated Poisson and lifetime distributions is introduced. A specialization is paved to a new three-parameter distribution, called doubly Poisson-exponential distribution, which may represent the lifetime of units connected in a series-parallel system. The new distribution can be obtained by compounding two zero truncated Poisson distributions with an exponential distribution. Among its motivations is that its hazard rate function can take different shapes such as decreasing, increasing and upside-down bathtub depending on the values of its parameters. Several properties of the new distribution are discussed. Based on progressive type-II censoring, six estimation methods [maximum likelihood, moments, least squares, weighted least squares and Bayes (under linear-exponential and general entropy loss functions) estimations] are used to estimate the involved parameters. The performance of these methods is investigated through a simulation study. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. In addition, confidence intervals, symmetric credible intervals and highest posterior density credible intervals of the parameters are obtained. Finally, an application to a real data set is used to compare the new distribution with other five distributions.

Bayes inference using half-logistic generated Weibull model based on type-II censoring

ABSTRACT A new distribution, called half-logistic generated Weibull distribution (HLGWD), is obtained by composing a half-logistic cumulative distribution function H with a function , where G(x) is Weibull, such that H(η(x)) is a survival function. Some properties of the new distribution are presented. A real data set is analyzed using the generated class of distributions which shows that the HLGWD can be used quite effectively in analyzing real lifetime data. Bayes estimation and one-sample Bayes prediction of future observables from the HLGWD are obtained and computed.