Mohamed A. W. Mahmoud

Inferences of the Lifetime Performance Index with Lomax Distribution Based on Progressive Type-II Censored Data

Abstract Effective management and the assessment of quality performance of products is important in modern enterprises. Often, the business performance is measured using the lifetime performance index CL to evaluate the potential of a process, where L is a lower specification limit. In this paper the maximum likelihood estimator (MLE) of CL is derived based on progressive Type II sampling and assuming the Lomax distribution. The MLE of CL is then utilized to develop a new hypothesis testing procedure for given value of L. Moreover, we develop the Bayes estimator of CL assuming the conjugate prior distribution and applying the squared-error loss function. The Bayes estimator of CL is then utilized to develop a credible interval again for given L. Finally, we propose a Bayesian test to assess the lifetime performance of products and give two examples and a Monte Carlo simulation to assess and compare the two ML-approach with the Bayes-approach with respect to the lifetime performance index CL.

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.

Statistical inferences for new Weibull-Pareto distribution under an adaptive type-ii progressive censored data

Abstract In this paper, we obtain the maximum likelihood, Bayes and parametric bootstrap estimators for the parameters of a new Weibull-Pareto distribution (NWPD) and some lifetime indices such as reliability function S(t), failure rate h(t) function and coefficient of variation CV are obtained. The previous methods are studied in the case of an adaptive Type-II progressive censoring (Ada-T-II-Pro-C). Approximate confidence intervals (ACIs) of the unknown parameters are constructed based on the asymptotic normality of maximum likelihood estimators (MLEs). Bayes estimates and the symmetric credible intervals (CRIs) of the unknown quantities are calculated based on the Gibbs sampler within Metropolis– Hasting (M-H) algorithm procedure. The results of Bayes estimates are obtained under the consideration of the informative prior function with respect to the squared error loss (SEL) function. Two numerical examples are presented to illustrate the proposed methods, one of them is a simulated example and the other is a real life example. Finally, the performance of different Bayes estimates are compared with maximum likelihood (ML) and two parametric bootstrap estimates, through a Monte Carlo simulation study.