Essam K. AL-Hussaini

Predicting observables from a general class of distributions

A general class of distributions is proposed to be the underlying population model from which observables are to be predicted using the Bayesian approach. This class of distributions includes, among others, the Weibull, compound Weibull (or three-parameter Burr-type XII), Pareto, beta, Gompertz and compound Gompertz distributions. A proper general prior density function is suggested and the predictive density functions are obtained in the one- and two-sample cases. The informative sample is assumed to be a type II censored sample. Illustrative examples of Weibull (α,β), Burr-type XII (α,β), and Pareto (α,β) distributions are given and compared with the results obtained by previous researchers.

Bayesian estimation of the parameters,reliability and failure rate functions of the burr type xii failure model

Based on a type-2 censored sample of the life times from a two parameter Burr type-XII failure time model, the Bayes estimates of the two (unknown) parameters, the reliability and failure rate functions are obtained by using Bayes approximation form due to Lindley (1980). The estimated risks of the Bayes estimates are computed and compared with the corresponding estimated risks of the maximum likelihood estimates.

On Bayesian interval prediction of future records

Based on the one-sample scheme, Bayesian prediction bounds for the sth future record value are obtained. All of the informative and future observations are assumed to be obtained from a general class of distributions which includes the Weibull, compound Weibull, Pareto, beta, Gompertz, compound Gompertz among other distributions. The prior belief of the experimenter is measured by a proper general conjugate prior which was suggested by AL-Hussaini (1999b).

Bayesian prediction bounds for the burr type XII failure model

In this paper, the problem of predicting the future observations from the Burr type XII distribution, based on the past observations having the same distribution, is considered from a Bayesian approach One sample and a ser ies of M+l samples techniques are used in this article based on a bivariate prior density suggested by AL.-Hussaini and Jaheen (1992). Numerical examples are used to illustrate the procedures.

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.

Bayesian prediction bounds for the Burr type XII distribution in the presence of outliers

Abstract Bayesian prediction bounds for some order statistics of future observations from the Burr (c, k) distribution are obtained in the presence of single outlier arising from different members of the same family of distributions. Single outliers of types kk0 and k + k0 are considered and a bivariate prior density for c and k as suggested by AL-Hussaini and Jaheen (J. Statist. Comp. and Simulation 41 (1992)) is used.

Bayesian one-sample prediction of future observations under Pareto distribution

Suppose that the length of time in years for which a business operates until failure has a Pareto distribution. Let t 1 < t 2 < ··· < t r denote the survival lifetimes of the first r of a random sample of n businesses. Bayesian predictions are to be made on the ordered failure times of the remaining (n − r) businesses, using the conditional probability function. Numerical examples are given to illustrate our results.

Approximate Bayes estimators applied to the Burr model

This paper extends the results of AL-Hussaini and Jaheen (1992) and develops approximate Bayes estimators of the two (unknown) parameters, reliability and failure rate functions of the Burr type XII failure model by using the method of Tierney and Kadane (1986) based on type-2 censored samples. Comparisons are made between those estimators and their corresponding Bayes estimators obtained by using the method of Lindley (1980) together with the maximum likelihood estimators based on Monte Carlo simulation study.

Accelerated life tests under finite mixture models

In this paper, the failure time of a device is observed under a higher stress subjected to a general class of stress-response model, when its distribution is a mixture of k components each of which represents a different cause of failure. The problem is studied when each of the components belongs to 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. On the basis of the censored data, the maximum likelihood estimates of the unknown parameters involved under the general stress-response model are obtained. A special attention is paid to the power rule model applied to mixtures of two Weibull components. Mixtures of two exponentials, Rayleigh and Weibull components models are used as illustrative examples.

Bayesian prediction based on finite mixtures of lomax components model and type i censoring

This paper is concerned with the problem of obtaining Bayesian prediction bounds for future observations based on a type I censored sample from a nonhomogerieous population having a distribution which is a mixture of two Lomax components. A numerical example is given to illustrate our results.