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28 pages, 835 KB

Open AccessArticle

Progressive First-Failure Censoring in Reliability Analysis: Inference for a New Weibull–Pareto Distribution

by Rashad M. EL-Sagheer and Mahmoud M. Ramadan

Mathematics 2025, 13(15), 2377; https://doi.org/10.3390/math13152377 - 24 Jul 2025

Viewed by 671

This paper explores statistical techniques for estimating unknown lifetime parameters using data from a progressive first-failure censoring scheme. The failure times are modeled with a new Weibull–Pareto distribution. Maximum likelihood estimators are derived for the model parameters, as well as for the survival and hazard rate functions, although these estimators do not have explicit closed-form solutions. The Newton–Raphson algorithm is employed for the numerical computation of these estimates. Confidence intervals for the parameters are approximated based on the asymptotic normality of the maximum likelihood estimators. The Fisher information matrix is calculated using the missing information principle, and the delta technique is applied to approximate confidence intervals for the survival and hazard rate functions. Bayesian estimators are developed under squared error, linear exponential, and general entropy loss functions, assuming independent gamma priors. Markov chain Monte Carlo sampling is used to obtain Bayesian point estimates and the highest posterior density credible intervals for the parameters and reliability measures. Finally, the proposed methods are demonstrated through the analysis of a real dataset. Full article

(This article belongs to the Section D1: Probability and Statistics)

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22 pages, 479 KB

Open AccessArticle

Classical and Bayesian Inference for a Progressive First-Failure Censored Left-Truncated Normal Distribution

by Yuxin Cai and Wenhao Gui

Symmetry 2021, 13(3), 490; https://doi.org/10.3390/sym13030490 - 16 Mar 2021

Cited by 10 | Viewed by 3073

Abstract

Point and interval estimations are taken into account for a progressive first-failure censored left-truncated normal distribution in this paper. First, we derive the estimators for parameters on account of the maximum likelihood principle. Subsequently, we construct the asymptotic confidence intervals based on these estimates and the log-transformed estimates using the asymptotic normality of maximum likelihood estimators. Meanwhile, bootstrap methods are also proposed for the construction of confidence intervals. As for Bayesian estimation, we implement the Lindley approximation method to determine the Bayesian estimates under not only symmetric loss function but also asymmetric loss functions. The importance sampling procedure is applied at the same time, and the highest posterior density (HPD) credible intervals are established in this procedure. The efficiencies of classical statistical and Bayesian inference methods are evaluated through numerous simulations. We conclude that the Bayes estimates given by Lindley approximation under Linex loss function are highly recommended and HPD interval possesses the narrowest interval length among the proposed intervals. Ultimately, we introduce an authentic dataset describing the tensile strength of 50mm carbon fibers as an illustrative sample. Full article

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28 pages, 640 KB

Open AccessArticle

Statistical Inference of the Lifetime Performance Index with the Log-Logistic Distribution Based on Progressive First-Failure-Censored Data

by Ying Xie and Wenhao Gui

Symmetry 2020, 12(6), 937; https://doi.org/10.3390/sym12060937 - 3 Jun 2020

Cited by 13 | Viewed by 3035

Abstract

Estimating the accurate evaluation of product lifetime performance has always been a hot topic in manufacturing industry. This paper, based on the lifetime performance index, focuses on its evaluation when a lower specification limit is given. The progressive first-failure-censored data we discuss have a common log-logistic distribution. Both Bayesian and non-Bayesian method are studied. Bayes estimator of the parameters of the log-logistic distribution and the lifetime performance index are obtained using both the Lindley approximation and Monte Carlo Markov Chain methods under symmetric and asymmetric loss functions. As for interval estimation, we apply the maximum likelihood estimator to construct the asymptotic confidence intervals and the Metropolis–Hastings algorithm to establish the highest posterior density credible intervals. Moreover, we analyze a real data set for demonstrative purposes. In addition, different criteria for deciding the optimal censoring scheme have been studied. Full article

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29 pages, 636 KB

Open AccessArticle

Parameter and Reliability Inferences of Inverted Exponentiated Half-Logistic Distribution under the Progressive First-Failure Censoring

by Fengshi Zhang and Wenhao Gui

Mathematics 2020, 8(5), 708; https://doi.org/10.3390/math8050708 - 3 May 2020

Cited by 8 | Viewed by 2687

Abstract

Using progressive first-failure censored samples, we mainly study the inferences of the unknown parameters and the reliability and failure functions of the Inverted Exponentiated Half-Logistic distribution. The progressive first-failure censoring is an extension and improvement of progressive censoring, which is of great significance in the field of lifetime research. Besides maximum likelihood estimation, we use Bayesian estimation under unbalanced and balanced losses: General Entropy loss function, Squared Error loss function and Linex loss function. Approximate explicit expression of Bayesian estimation is given using Lindley approximation method for point estimation and Metropolis-Hastings method for point and interval estimation. Bayesian credible intervals and asymptotic confidence intervals are derived in the form of average length and coverage probability. To show the research effects, a simulation study and practical data analysis are carried out. Finally, we discuss the optimal censoring mode under four different criteria. Full article

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24 pages, 581 KB

Open AccessArticle

Bayesian Inference of δ = P(X < Y) for Burr Type XII Distribution Based on Progressively First Failure-Censored Samples

by Jessie Marie Byrnes, Yu-Jau Lin, Tzong-Ru Tsai and Yuhlong Lio

Mathematics 2019, 7(9), 794; https://doi.org/10.3390/math7090794 - 1 Sep 2019

Cited by 9 | Viewed by 3544

Abstract

Let X and Y follow two independent Burr type XII distributions and

δ = P (X < Y)

. If X is the stress that is applied to a certain component and Y is the strength to sustain the stress, then [...] Read more.

Let X and Y follow two independent Burr type XII distributions and

δ = P (X < Y)

. If X is the stress that is applied to a certain component and Y is the strength to sustain the stress, then

δ

is called the stress–strength parameter. In this study, The Bayes estimator of

δ

is investigated based on a progressively first failure-censored sample. Because of computation complexity and no closed form for the estimator as well as posterior distributions, the Markov Chain Monte Carlo procedure using the Metropolis–Hastings algorithm via Gibbs sampling is built to collect a random sample of

δ

via the joint distribution of the progressively first failure-censored sample and random parameters and the empirical distribution of this collected sample is used to estimate the posterior distribution of

δ

. Then, the Bayes estimates of

δ

using the square error, absolute error, and linear exponential error loss functions are obtained and the credible interval of

δ

is constructed using the empirical distribution. An intensive simulation study is conducted to investigate the performance of these three types of Bayes estimates and the coverage probabilities and average lengths of the credible interval of

δ

. Moreover, the performance of the Bayes estimates is compared with the maximum likelihood estimates. The Internet of Things and a numerical example about the miles-to-failure of vehicle components for reliability evaluation are provided for application purposes. Full article

(This article belongs to the Special Issue Statistics and Modeling in Reliability Engineering)

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