Ahmed A. Soliman

Modified Weibull model: A Bayes study using MCMC approach based on progressive censoring data

In this paper, we investigate the problem of point and interval estimations for the modified Weibull distribution (MWD) using progressively type-II censored sample. The maximum likelihood (ML), Bayes, and parametric bootstrap methods are used for estimating the unknown parameters as well as some lifetime parameters (reliability and hazard functions). Also, we propose to apply Markov chain Monte Carlo (MCMC) technique to carry out a Bayesian estimation procedure. Bayes estimates and the credible intervals are obtained under the assumptions of informative and noninformative priors. The results of Bayes method are obtained under both the balanced squared error loss (bSEL) and balanced linear-exponential (bLINEX) loss. We show that these loss functions are more general, which include the MLE and both symmetric and asymmetric Bayes estimates as special cases. Finally, Two real data sets have been analyzed for illustrative purposes.

Estimation of the parameters of life for Gompertz distribution using progressive first-failure censored data

Bayes and frequentist estimators are obtained for the two-parameter Gompertz distribution (GD), as well as the reliability and hazard rate functions, using progressive first-failure censoring plan. We have examined Bayes estimates under symmetric and asymmetric loss functions. We show that the Bayes estimates relative to asymmetric loss function includes the maximum likelihood estimate (MLE) and other Bayes estimates as special cases. This is done using the conjugate prior for the scale parameter and discrete prior for the shape parameter. It has been seen that the Bayes estimators are obtained in closed form. Also, based on this new censoring scheme, exact and approximate confidence intervals as well as exact confidence region for the parameters of GD are developed. A practical example using simulated data set was used for illustration. Finally, to assess the performance of the proposed estimators, numerical results using Monte Carlo simulation study were reported.

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 Kuş [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.

Reliability estimation in stress–strength models: an MCMC approach

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.

Estimating the Pareto parameters under progressive censoring data for constant-partially accelerated life tests

Accelerated life testing is widely used in product life testing experiments since it provides significant reduction in time and cost of testing. In this paper, assuming that the lifetime of items under use condition follow the two-parameter Pareto distribution of the second kind, partially accelerated life tests based on progressively Type-II censored samples are considered. The likelihood equations of the model parameters and the acceleration factor are reduced to a single nonlinear equation to be solved numerically to obtain the maximum-likelihood estimates (MLEs). Based on normal approximation to the asymptotic distribution of MLEs, the approximate confidence intervals (ACIs) for the parameters are derived. Two bootstrap CIs are also proposed. The classical Bayes estimates cannot be obtained in explicit form, so we propose to apply Markov chain Monte Carlo method to tackle this problem, which allows us to construct the credible interval of the involved parameters. Analysis of a simulated data set has also been presented for illustrative purposes. Finally, a Monte Carlo simulation study is carried out to investigate the precision of the Bayes estimates with MLEs and to compare the performance of different corresponding CIs considered.

Estimation of the coefficient of variation for non-normal model using progressive first-failure-censoring data

The coefficient of variation (CV) is extensively used in many areas of applied statistics including quality control and sampling. It is regarded as a measure of stability or uncertainty, and can indicate the relative dispersion of data in the population to the population mean. In this article, based on progressive first-failure-censored data, we study the behavior of the CV of a random variable that follows a Burr-XII distribution. Specifically, we compute the maximum likelihood estimations and the confidence intervals of CV based on the observed Fisher information matrix using asymptotic distribution of the maximum likelihood estimator and also by using the bootstrapping technique. In addition, we propose to apply Markov Chain Monte Carlo techniques to tackle this problem, which allows us to construct the credible intervals. A numerical example based on real data is presented to illustrate the implementation of the proposed procedure. Finally, Monte Carlo simulations are performed to observe the behavior of the proposed methods.

Markov Chain Monte Carlo to Study the Estimation of the Coefficient of Variation

The coefficient of variation (CV ) of a population is defined as the ratio of the population standard deviation to the population mean. It is regarded as a measure of stability or uncertainty, and can indicate the relative dispersion of data in the population to the population mean. In this article, based on the upper record values, we study the behavior of the CV of a random variable that follows a Lomax distribution. Specifically, we compute the maximum likelihood estimations (MLEs) and the confidence intervals of CV based on the observed Fisher information matrix using asymptotic distribution of the maximum likelihood estimator and also by using the bootstrapping technique. In addition, we propose to apply Markov Chain Monte Carlo (MCMC) techniques to tackle this problem, which allows us to construct the credible intervals. A numerical example based on a real data is presented to illustrate the implementation of the proposed procedure. Finally, Monte Carlo simulations are performed to observe the behavior of the proposed methods.

Bayesian MCMC inference for the Gompertz distribution based on progressive first-failure censoring data

This article deals with the problem of estimating parameters of the Gompertz distribution (GD) based on progressive first-failure censored data using Bayesian and non-Bayesian approaches. The two-sample prediction problem is considered to derive Bayesian prediction bounds for both future order statistics and future record values based on progressive first failure censored informative samples from GD. The sampling schemes such as, first-failure censoring, progressive type II censoring, type II censoring and complete sample can be obtained as special cases of the progressive first-failure censored scheme. Markov chain Monte Carlo (MCMC) method with Gibbs sampling procedure is used to compute the Bayes estimates and also to construct the corresponding credible intervals of the parameters. A simulation study has been conducted in order to compare the proposed Bayes estimators with the maximum likelihood estimators MLE. Finally, some numerical computations with real data set are presented for illustrating all ...

Bayesian and non-Bayesian inferences of the Burr-XII distribution for progressive first-failure censored data

SummaryIn this paper, based on a new type of censoring scheme called a progressive first-failure censored, the maximum likelihood (ML) and the Bayes estimators for the two unknown parameters of the Burr type XII distribution are derived. This type of censoring contains as special cases various types of censoring schemes used in the literature. When the two parameters are unknown, the Bayes estimators can not be obtained in explicit forms. We use Lindley’s approximation to compute the Bayes estimates and the Gibbs sampling procedure to calculate the credible intervals. A Bayesian approach using Markov Chain Monte Carlo (MCMC) techniques to generate from the posterior distributions and in turn computing the Bayes estimators is developed. Point estimation and confidence intervals based on maximum likelihood and bootstrap methods are also proposed. The approximate Bayes estimators have been obtained under the assumptions of informative and non-informative priors. A numerical example using real data set is provided to illustrate the proposed methods. Finally, the maximum likelihood and different Bayes estimators are compared via a Monte Carlo simulation study.