Planning Step-Stress Life Tests for the Generalized Rayleigh Distribution under Progressive Type-II Censoring with Binomial Removals

In this article, both the parameter estimation and optimal design problems of step-stress partially accelerated life test units whose lifetimes follow the generalized Rayleigh distribution are considered under progressive type-II censoring scheme with binomial removals. The maximum likelihood estimators of the scale and shape parameters as well as the acceleration factor are obtained. The concert of the estimators is assessed. In addition, approximate confidence intervals of the model parameters are constructed and their coverage probabilities are computed. Moreover, optimum test plans are also developed to improve/guarantee the quality of the statistical inference. Finally, simulation studies and a numerical example are provided for illustrative purposes.Рассмотрены параметр оценки и оптимальное проектирование частично ускоренных испытаний на долговечность при ступенчатой нагрузке на основе обобщенного рэлеевского распределения при прогрессивном цензурировании типа II с биномиальными выборками. В качестве фактора ускорения используются максимальные оценки вероятности параметров масштаба и формы, которые согласуются между собой. Построены приближенные доверительные интервалы параметров модели и рассчитаны границы вероятности. Разработаны оптимальные планы испытаний для улучшения статистического анализа. Предложены результаты моделирования и числовой пример.Розглянуто параметр оцінки і оптимальне проектування частково прискорених випробувань на довговічність при ступеневому навантаженні на основі узагальненого релеївського розподілу при прогресивному цензуруванні типу ІІ з біноміальними виборками. Як фактор прискорення використовуються максимальні оцінки імовірності параметрів масштабу і форми, які узгоджуються між собою. Побудовано наближені довірчі інтервали параметрів моделі і розраховано границі імовірності. Розроблено оптимальні плани випробувань із метою покращання статистичного аналізу. Запропоновано результати моделювання і числовий приклад

Inference and optimal design for the k-level step-stress accelerated life test based on progressive Type-I interval censored power Rayleigh data

In this paper, a new generalization of the one parameter Rayleigh distribution called the Power Rayleigh (PRD) was employed to model the life of the tested units in the step-stress accelerated life test. Under progressive Type-I interval censored data, the cumulative exposure distribution was considered to formulate the life model, assuming the scale parameter of PRD has the inverse power function at each stress level. Point estimates of the model parameters were obtained via the maximum likelihood estimation method, while interval estimates were obtained using the asymptotic normality of the derived estimators and the bootstrap resampling method. An extensive simulation study of $k = 4$ levels of stress in different combinations of the life test under different progressive censoring schemes was conducted to investigate the performance of the obtained point and interval estimates. Simulation results indicated that point estimates of the model parameters are closest to their initial true values and have relatively small mean squared errors. Accordingly, the interval estimates have small lengths and their coverage probabilities are almost convergent to the 95% significance level. Based on the Fisher information matrix, the D-optimality and the A-optimality criteria are implemented to determine the optimal design of the life test by obtaining the optimum inspection times and optimum stress levels that improve the estimation procedures and give more efficient estimates of the model parameters. Finally, the developed inferential procedures were also applied to a real dataset

Optimal Test Plan of Step Stress Partially Accelerated Life Testing for Alpha Power Inverse Weibull Distribution under Adaptive Progressive Hybrid Censored Data and Different Loss Functions

Accelerated life tests are used to explore the lifetime of extremely reliable items by subjecting them to elevated stress levels from stressors to cause early failures, such as temperature, voltage, pressure, and so on. The alpha power inverse Weibull (APIW) distribution is of great significance and practical applications due to its appealing characteristics, such as its flexibilities in the probability density function and the hazard rate function. We analyze the step stress partially accelerated life testing model with samples from the APIW distribution under adaptive type II progressively hybrid censoring. We first obtain the maximum likelihood estimates and two types of approximate confidence intervals of the distributional parameters and then derive Bayes estimates of the unknownparameters under different loss functions. Furthermore, we analyze three probable optimum test techniques for identifying the best censoring under different optimality criteria methods. We conduct simulation studies to assess the finite sample performance of the proposed methodology. Finally, we provide a real data example to further demonstrate the proposed technique

Optimum Failure-Censored Step-Stress Life Test Plans for the Lomax Distribution

Рассматриваются частично ускоренные ресурсные испытания при пошаговом изменении напряжений, при которых постулируется, что время до разрушения характеризуется распределением Ломакса при цензурировании разрушения. Получены показатели максимальной вероятности параметров данной модели и соответствующие среднеквадратичные отклонения, а также рассчитаны доверительные интервалы параметров с соответствующими вероятностями покрытия. Изучены оптимальные варианты проведения ресурсных испытаний. Для верификации полученных теоретических результатов выполнено численное моделирование тестовых задач.Розглядаються частково прискорені ресурсні дослідження при покроковій зміні напружень, за яких постулюється, що час до руйнування характеризується розподілом Ломакса при цензуруванні руйнування. Отримано показники максимальної імовірності параметрів даної моделі і відповідні середньоквадратичні відхилення та розраховано довірчі інтервали параметрів із відповідними імовірностями покриття. Вивчено оптимальні варіанти проведення ресурсних випробувань. Для верифікації отриманих теоретичних результатів виконано чисельне моделювання тестових задач

Designing Advanced Reliability Testing Mathematical Model for Modern Products

The modern era is the age of science, technology and at the same time it is the age of competition. The advancement of new technology and increased global competition have emphasized the importance of product strength and reliability estimation. As a result, producers and manufacturers must now verify the strength and reliability of their products prior to releasing them to the market. In the past, reliability data analysis was a critical tool for this purpose. Traditionally, reliability data analysis entails quantifying these life characteristics through the examination of failure data. However, in many situations, obtaining such failure data has been extremely difficult, if not impossible, due to the length of time between designing and releasing a product, and the difficulty of designing a product that will last a long period due to its continuous use and operation. Faced with this challenge, reliability statisticians developed a technique called Accelerated Reliability Testing to rapidly determine the reliability and life characteristics of products. This technique increases product reliability and identifies when and how a product will fail in its intended environment. In the present work, we plan to investigate these mathematical reliability models to determine the costs associated with the various product guarantees. If component lifetimes follow the power-function distribution, the problem is examined under increasing stress using percent failure censoring. The method is referred as a process that applies accelerated testing to estimate the cost of age-replacement for goods sold under warranty. Additionally, a mathematical illustration is presented to illustrate the results

Bayesian Sequential Design Based on Dual Objectives for Accelerated Life Tests

Traditional accelerated life test plans are typically based on optimizing the C-optimality for minimizing the variance of an interested quantile of the lifetime distribution. The traditional methods rely on some specified planning values for the model parameters, which are usually unknown prior to the actual tests. The ambiguity of the specified parameters can lead to suboptimal designs for optimizing the intended reliability performance. In this paper, we propose a sequential design strategy for life test plans based on considering dual objectives. In the early stage of the sequential experiment, we suggest to allocate more design locations based on optimizing the D-optimality to quickly gain precision in the estimated model parameters. In the later stage of the experiment, we can allocate more samples based on optimizing the C-optimality to maximize the precision of the estimated quantile of the lifetime distribution. We compare the proposed sequential design strategy with existing test plans considering only a single criterion and illustrate the new method with an example on fatigue testing of polymer composites.Comment: 17 page

Accelerated life test method for the doubly truncated Burr type XII distribution

[[abstract]]The Burr type XII (BurrXII) distribution is very flexible for modeling and has earned much attention in the past few decades. In this study, the maximum likelihood estimation method and two Bayesian estimation procedures are investigated based on constant-stress accelerated life test (ALT) samples, which are obtained from the doubly truncated three-parameter BurrXII distribution. Because computational difficulty occurs for maximum likelihood estimation method, two Bayesian procedures are suggested to estimate model parameters and lifetime quantiles under the normal use condition. A Markov Chain Monte Carlo approach using the Metropolis–Hastings algorithm via Gibbs sampling is built to obtain Bayes estimators of the model parameters and to construct credible intervals. The proposed Bayesian estimation procedures are simple for practical use, and the obtained Bayes estimates are reliable for evaluating the reliability of lifetime products based on ALT samples. Monte Carlo simulations were conducted to evaluate the performance of these two Bayesian estimation procedures. Simulation results show that the second Bayesian estimation procedure outperforms the first Bayesian estimation procedure in terms of bias and mean squared error when users do not have sufficient knowledge to set up hyperparameters in the prior distributions. Finally, a numerical example about oil-well pumps is used for illustration.[[notice]]補正完

Finding appropriate loss distributions to insurance data Case study of Kenya (2010-2014)

A Research project submitted in partial fulfillment of the requirements for the degree of Bachelor of Business Science in Actuarial Science at Strathmore UniversityObtaining the total amount of claims for a specific period is a vital part of the daily work of insurance companies. This will help in various ways the management in running the company (Jouravlev, 2009). For instance, the insurance company will be able to calculate the premium for a type of policy by the use of the claim experience. Moreover, it will be able to reserve a certain amount of money to cover the cost of future claims. Premium computation and Reserving are not the only reasons for which loss distributions are needed. Loss distributions are also utilised in reviewing reinsurance arrangements and also in testing for solvency. This explicitly highlights the importance of loss distribution in the insurance industry. This paper therefore aims to determine the most suitable loss distributions for various sort of insurance contracts being general or life insurance in the Kenyan market industry. The following distributions will be compared: the exponential distribution, the Pareto distribution, the Generalised Pareto distribution, the lognormal distribution, the Weibull distribution & the Burr distribution. We will see how these distributions can be tailored in order to suit the observed data. Afterwards, a test of goodness-of-fit will be used to determine the level of robustness of the distribution in fitting the given data. The loss distributions will also be used in order the probabilities of future events happening

Statistical inference for the Power Rayleigh distribution based on adaptive progressive Type-II censored data

The Power Rayleigh distribution (PRD) is a new extension of the standard one-parameter Rayleigh distribution. To employ this distribution as a life model in the analysis of reliability and survival data, we focused on the statistical inference for the parameters of the PRD under the adaptive Type-II censored scheme. Point and interval estimates for the model parameters and the corresponding reliability function at a given time are obtained using likelihood, Bootstrap and Bayesian estimation methods. A simulation study is conducted in different settings of the life testing experiment to compare and evaluate the performance of the estimates obtained. In addition, the estimation procedure is also investigated in real lifetimes data. The results indicated that the obtained estimates gave an accurate and efficient estimation of the model parameters. The Bootstrap estimates are better than the estimates obtained by the likelihood estimation approach, and estimates obtained using the Markov Chain Monte Carlo method by the Bayesian approach under both the squared error and the general entropy loss functions have priority over other point and interval estimates. Under the adaptive Type-II censoring scheme, concluding results confirmed that the PRD can be effectively used to model the lifetimes in survival and reliability analysis