Ali Akbar Jafari

Process capability indices are essential tools for assessing product quality, offering insights into how well a product meets specified standards and customer expectations. This paper explores the lifetime performance index under the assumption that product lifetimes follow a Weibull distribution. We first develop six generalized approaches using the concept of a generalized pivotal quantity and incorporate properties of progressively first‐failure‐censored samples, as well as maximum likelihood and weighted least squares estimation for the Weibull distribution. Additionally, we introduce a Bayesian approach for inference on the lifetime performance parameter and employ Markov chain Monte Carlo simulations to compute posterior estimates and construct credible intervals. Furthermore, we extend another Bayesian approach that uses a different strategy for specifying the prior distribution of parameters—a transition from progressive Type II censoring to progressive first‐failure censoring. We analyze the properties of confidence intervals for the performance index through Monte Carlo simulations, comparing the results of a large‐sample method based on maximum likelihood estimation and the extended Bayesian approach with the seven novel approaches. The behavior of the confidence intervals for the parameter is illustrated through two examples.

The notations used in equations (6) and (7) require further clarification, as they are currently ambiguous. Specifically, the interpretation of the moments mentioned in these equations should be explained in greater detail to ensure the reader unders...

Performance capability indices are valuable measures to evaluate the quality of a product. In this paper, we consider inference on a lifetime performance index when the product’s lifetime follows a gamma distribution with unknown parameters. The bias and mean square error of the maximum likelihood estimator and other proposed estimators are compared. Also, an asymptotic confidence interval using the maximum likelihood estimator, thirteen bootstrap confidence intervals, and four generalized pivotal quantities are derived for the performance capability index. A Monte Carlo simulation is provided to investigate the accuracy of expected lengths and coverage probabilities of the confidence intervals. An actual data set is used to illustrate the estimators and confidence intervals.

In this paper, a measure of extropy is obtained for concomitants of m-generalized order statistics in the Morgenstern family. The cumulative residual extropy (CREX) and negative cumulative extropy (NCEX) are presented for therth concomitant of m-generalized order statistics. In addition, the problem of estimating the CREX and NCEX is studied utilizing the empirical method in concomitants of m-generalized order statistics. Some applications of these results are given for the concomitants of order statistics and record values.

Stress-strength reliability is a measure to compare the lifetimes of two systems. It is inferred for the two-parameter exponential distribution using generalized order statistics first without constraint on the location and scale parameters, second when the scale parameters are equal. A generalized confidence interval, bootstrap confidence intervals, a Bayesian interval, and a highest posterior density interval are computed for the stress-strength parameter. A Monte Carlo simulation shows that generalized confidence intervals provide more accurate average lengths of confidence intervals and higher probabilities to contain the true value of the parameter. Application: Confidence intervals for the time to remission of 20 leukemic patients treated with one of two drugs are approximately the same in most generalized statistical models. In addition, the time to remission for patients with the first drug is tested to be shorter than for patients with the second drug.