Theoretical framework and inference for fitting extreme data through the modified Weibull distribution in a first-failure censored progressive approach.

The importance of biomedical physical data is underscored by its crucial role in advancing our comprehension of human health, unraveling the mechanisms underlying diseases, and facilitating the development of innovative medical treatments and interventions. This data serves as a fundamental resource, empowering researchers, healthcare professionals, and scientists to make informed decisions, pioneer research, and ultimately enhance global healthcare quality and individual well-being. It forms a cornerstone in the ongoing pursuit of medical progress and improved healthcare outcomes. This article aims to tackle challenges in estimating unknown parameters and reliability measures related to the modified Weibull distribution when applied to censored progressive biomedical data from the initial failure occurrence. In this context, the article proposes both classical and Bayesian techniques to derive estimates for unknown parameters, survival, and failure rate functions. Bayesian estimates are computed considering both asymmetric and symmetric loss functions. The Markov chain Monte Carlo method is employed to obtain these Bayesian estimates and their corresponding highest posterior density credible intervals. Due to the inherent complexity of these estimators, which cannot be theoretically compared, a simulation study is conducted to evaluate the performance of various estimation procedures. Additionally, a range of optimization criteria is utilized to identify the most effective progressive control strategies. Lastly, the article presents a medical application to illustrate the effectiveness of the proposed estimators. Numerical findings indicate that Bayesian estimates outperform other estimation methods by achieving minimal root mean square errors and narrower interval lengths.

One- and Two-Sample Predictions Based on Progressively Type-II Censored Carbon Fibres Data Utilizing a Probability Model.

New Weibull-Pareto distribution is a significant and practical continuous lifetime distribution, which plays an important role in reliability engineering and analysis of some physical properties of chemical compounds such as polymers and carbon fibres. In this paper, we construct the predictive interval of unobserved units in the same sample (one sample prediction) and the future sample based on the current sample (two-sample prediction). The used samples are generated from new Weibull-Pareto distribution due to a progressive type-II censoring scheme. Bayesian and maximum likelihood approaches are implemented to the prediction problems. In the Bayesian approach, it is not easy to simplify the predictive posterior density function in a closed form, so we use the generated Markov chain Monte Carlo samples from the Metropolis-Hastings technique with Gibbs sampling. Moreover, the predictive interval of future upper-order statistics is reported. Finally, to demonstrate the proposed methodology, both simulated data and real-life data of carbon fibres examples are considered to show the applicabilities of the proposed methods.

Asymmetric Power Hazard Distribution for COVID-19 Mortality Rate under Adaptive Type-II Progressive Censoring: Theory and Inferences.

This article investigates the estimation of the parameters for power hazard function distribution and some lifetime indices such as reliability function, hazard rate function, and coefficient of variation based on adaptive Type-II progressive censoring. From the perspective of frequentism, we derive the point estimations through the method of maximum likelihood estimation. Besides, delta method is implemented to construct the variances of the reliability characteristics. Markov chain Monte Carlo techniques are proposed to construct the Bayes estimates. To this end, the results of the Bayes estimates are obtained under squared error and linear exponential loss functions. Also, the corresponding credible intervals are constructed. A simulation study is utilized to assay the performance of the proposed methods. Finally, a real data set of COVID-19 mortality rate is analyzed to validate the introduced inference methods.

Inferences for Stress-Strength Reliability Model in the Presence of Partially Accelerated Life Test to Its Strength Variable.

We focus on estimating the stress-strength reliability model when the strength variable is subjected to the step-stress partially accelerated life test. Based on the assumption that both stress and strength random variables follow Weibull distribution with a common first shape parameter, the inferences for this reliability system are constructed. The maximum likelihood, two parametric bootstraps, and Bayes estimates are obtained. Moreover, approximate confidence intervals, asymptotic variance-covariance matrix, and highest posterior density credible intervals are derived. A simulation study and application to real-life data are conducted to compare the proposed estimation methods developed here and also check the accuracy of the results.

Assessing the Lifetime Performance Index with Digital Inferences of Power Hazard Function Distribution Using Progressive Type-II Censoring Scheme.

This paper deals with estimating the lifetime performance index. The maximum likelihood (ML) and Bayesian estimators for lifetime performance index C L X where L X is the lower specification limit are derived based on progressive type-II censored (Prog-Type-II-C) sample from two-parameter power hazard function distribution (PHFD). Knowing the lower specification limit, the MLE of C L X is applied to construct a new hypothesis testing procedure. Bayesian estimator of C L X is also utilized to develop a credible interval. Also, the relationship between the C L X and the conforming rate of products is investigated. Moreover, the Bayesian test to evaluate the lifetime performance of units is proposed. A simulation study and illustrative example based on a real dataset are discussed to evaluate the performance of the two tests.

Inferences for Weibull Fréchet Distribution Using a Bayesian and Non-Bayesian Methods on Gastric Cancer Survival Times.

In this article, based on progressively type-II censored schemes, the maximum likelihood, Bayes, and two parametric bootstrap methods are used for estimating the unknown parameters of the Weibull Fréchet distribution and some lifetime indices as reliability and hazard rate functions. Moreover, approximate confidence intervals and asymptotic variance-covariance matrix have been obtained. Markov chain Monte Carlo technique based on Gibbs sampler within Metropolis-Hasting algorithm is used to generate samples from the posterior density functions. Furthermore, Bayesian estimate is computed under both balanced square error loss and balanced linear exponential loss functions. Simulation results have been implemented to obtain the accuracy of the estimators. Finally, application on the survival times in years of a group of patients given chemotherapy and radiation treatment is presented for illustrating all the inferential procedures developed here.