Product Spacing of Stress–Strength under Progressive Hybrid Censored for Exponentiated-Gumbel Distribution

Maximum product spacing for stress–strength model based on progressive Type-II hybrid censored samples with different cases has been obtained. This paper deals with estimation of the stress strength reliability model R = P(Y < X) when the stress and strength are two independent exponentiated Gumbel distribution random variables with different shape parameters but having the same scale parameter. The stress–strength reliability model is estimated under progressive Type-II hybrid censoring samples. Two progressive Type-II hybrid censoring schemes were used, Case I: A sample size of stress is the equal sample size of strength, and same time of hybrid censoring, the product of spacing function under progressive Type-II hybrid censoring schemes. Case II: The sample size of stress is a different sample size of strength, in which the life-testing experiment with a progressive censoring scheme is terminated at a random time T ∈ (0,∞). The maximum likelihood estimation and maximum product spacing estimation methods under progressive Type-II hybrid censored samples for the stress strength model have been discussed. A comparison study with classical methods as the maximum likelihood estimation method is discussed. Furthermore, to compare the performance of various cases, Markov chain Monte Carlo simulation is conducted by using iterative procedures as Newton Raphson or conjugate-gradient procedures. Finally, two real datasets are analyzed for illustrative purposes, first data for the breaking strengths of jute fiber, and the second data for the waiting times before the service of the customers of two banks.     

Modelling Insurance Losses with a New Family of Heavy-Tailed Distributions

The actuaries always look for heavy-tailed distributions to model data relevant to business and actuarial risk issues. In this article, we introduce a new class of heavy-tailed distributions useful for modeling data in financial sciences. A specific sub-model form of our suggested family, named as a new extended heavy-tailed Weibull distribution is examined in detail. Some basic characterizations, including quantile function and raw moments have been derived. The estimates of the unknown parameters of the new model are obtained via the maximum likelihood estimation method. To judge the performance of the maximum likelihood estimators, a simulation analysis is performed in detail. Furthermore, some important actuarial measures such as value at risk and tail value at risk are also computed. A simulation study based on these actuarial measures is conducted to exhibit empirically that the proposed model is heavy-tailed. The usefulness of the proposed family is illustrated by means of an application to a heavy-tailed insurance loss data set. The practical application shows that the proposed model is more flexible and efficient than the other six competing models including (i) the two-parameter models Weibull, Lomax and Burr-XII distributions (ii) the three-parameter distributions Marshall-Olkin Weibull and exponentiated Weibull distributions, and (iii) a well-known four-parameter Kumaraswamy Weibull distribution.     

Maximum likelihood Estimation of Parameters in the Inverse Gaussian Distribution,With Unknown Origin

Maximum likelihood estimation is applied to the three-parameter Inverse Gaussian distribution, which includes an unknown shifted origin parameter. It is well known that for similar distributions in which the origin is unknown, such as the lognormal, gamma, and Weibull distributions, maximum likelihood estimation can break down. In these latter cases, the likelihood function is unbounded and this leads to inconsistent estimators or estimators not asymptotically normal. It is shown that in the case of the Inverse Gaussian distribution this difticulty does not arise. The likelihood remains bounded and maximum likelihood estimation yields a consistent estimator with the usual asymptotic normality properties. A simple iterative method is suggested for the estimation procedure. Numerical examples are given in which the estimates in the Inverse Gaussian model are compared with those of the lognormal and Weibull distributions.     

Inference for a geometric‐poisson‐Rayleigh distribution under progressive‐stress accelerated life tests based on type‐I progressive hybrid censoring with binomial removals

Based on failures of a parallel‐series system, a new distribution called geometric‐Poisson‐Rayleigh distribution is proposed. Some properties of the distribution are discussed. A real data set is used to compare the new distribution with other 6 distributions. The progressive‐stress accelerated life tests are considered when the lifetime of an item under use condition is assumed to follow the geometric‐Poisson‐Rayleigh distribution. It is assumed that the scale parameter of the geometric‐Poisson‐Rayleigh distribution satisfies the inverse power law such that the stress is a nonlinear increasing function of time and the cumulative exposure model for the effect of changing stress holds. Based on type‐I progressive hybrid censoring with binomial removals, the maximum likelihood and Bayes (using linear‐exponential and general entropy loss functions) estimation methods are considered to estimate the involved parameters. Some point predictors such as the maximum likelihood, conditional median, best unbiased, and Bayes point predictors for future order statistics are obtained. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. Finally, a simulation study is performed, and numerical computations are performed to compare the performance of the implemented methods of estimation and prediction.     

Generalized inverted exponential distribution under constant stress accelerated life test: Different estimation methods with application

Estimation of the parameters of generalized inverted exponential distribution is considered under constant stress accelerated life test. Besides the maximum likelihood method, nine different frequentist methods of estimation are used to estimate the unknown parameters. Moreover, the reliability function is estimated under use conditions based on different methods of estimation. We perform extensive simulation experiments to see the performance of the proposed estimators. As an illustration, a real data set is analyzed to demonstrate how the proposed methods may work in practice.     

Reliability analysis of exponentiated Poisson-exponential constant stress accelerated life test model

Accelerated life testing is an efficient tool frequently adopted for obtaining failure time data of test units in a lesser time period as compared to normal use conditions. We assume that the lifetime data of a product at constant level of stress follows an exponentiated Poisson-exponential distribution and the shape parameter of the model has a log-linear relationship with the stress level. Model parameters, the reliability function (RF), and the mean time to failure (MTTF) function under use conditions are estimated based on eight frequentist methods of estimation, namely, method of maximum likelihood, method of least square and weighted least square, method of maximum product of spacing, method of minimum spacing absolute-log distance, method of Cramér-von-Mises, method of Anderson–Darling, and Right-tail Anderson–Darling. The performance of the different estimation methods is evaluated in terms of their mean relative estimate and mean squared error using small and large sample sizes through a Monte Carlo simulation study. Finally, two accelerated life test data sets are considered and bootstrap confidence intervals for the unknown parameters, predicted shape parameter, predicted RF, and the MTTF at different stress levels, are obtained.     

Fitting polytomous Rasch models in SAS

The item parameters of a polytomous Rasch model can be estimated using marginal and conditional approaches. This paper describes how this can be done in SAS (V8.2) for three item parameter estimation procedures: marginal maximum likelihood estimation, conditional maximum likelihood estimation, and pairwise conditional estimation. The use of the procedures for extensions of the Rasch model is also discussed. The accuracy of the methods are evaluated using a simulation study.     

Reliability analysis of wind turbine subassemblies based on the 3-P Weibull model via an ergodic artificial bee colony algorithm

The Weibull distribution is the most widely used model for the reliability evaluation of wind turbine subassemblies. Considering the important role of the location parameter in the three-parameter (3-P) Weibull model and its rare application in wind turbines, this study conducted a reliability analysis of wind turbine subassemblies based on field data that obeyed the 3-P Weibull distribution model via maximum likelihood estimation (MLE). An improved ergodic artificial bee colony algorithm (ErgoABC) was proposed by introducing the chaos search theory, global best solution, and Lévy flights strategy into the classical artificial bee colony (ABC) algorithm to determine the maximum likelihood estimates of the Weibull distribution parameters. This was validated against simulation calculations and proved to be efficient for high-dimensional function optimization and parameter estimation of the 3-P Weibull distribution. Finally, reliability analyses of the wind turbine subassemblies based on different types of field failure data were conducted using ErgoABC. The results show that the 3-P Weibull model can reasonably evaluate the lifetime distribution of critical wind turbine subassemblies, such as generator slip rings and main shafts, on which the location parameter has a significant effect.     

A study of a storage reliability estimation problem

Storage reliability, which describes the failure or deterioration of items in a dormant state, is considered in this paper. The study presented here is focused on the estimation of the storage reliability after a certain amount of storage time. We start with simple non-parametric estimation of the current reliability and then study the problem of parametric estimation based on a simple Weibull distribution assumption. Both maximum likelihood estimation and graphical techniques are considered in this case. The study is useful for planning a storage environment and making a decision about the maximum length of storage. Furthermore, the information can be used in the design and improvement of products for which the storage is an important part of the product's life cycle. A numerical example is provided to enlighten the idea.     

Reliability inference for VGA adapter from dual suppliers based on contaminated type‐I interval‐censored data

Type‐I interval‐censoring scheme only documents the number of failed units within two prespecified consecutive exam times at the larger time point after putting all units on test at the initial time schedule. It is challenging to use the collected information from type‐I interval‐censoring scheme to evaluate the reliability of unit when not all admitted units are operated or tested at the same initial time and a majority of units are randomly selected to replace the failed test units at unrecorded time points. Moreover, the lifetime distribution of all pooled units from dual resources usually follows a mixture distribution. To overcome these two problems, a two‐stage inference process that consists of a data‐cleaning step and a parameter estimation step via either Markov chain Monte Carlo (MCMC) algorithm or profile likelihood method is proposed based on the contaminated type‐I interval‐censored sample from a mixture distribution with unknown proportion. An extensive simulation study is conducted under the mixture smallest extreme value distributions to evaluate the performance of the proposed method for a case study. Finally, the proposed methods are applied to the mixture lifetime distribution modeling of video graphics array adapters for the support of reliability decision.     

双参数Rayleigh分布环境因子的经验Bayes估计及应用

本文针对Rayleigh分布位置参数已知的情形,给出了Rayleigh分布环境因子的极大似然估计和经验Bayes估计,并将环境因子的估计结果应用于Rayleigh部件的可靠性评估,给出了该部件可靠度函数与失效率的估计。最后的随机模拟例子表明,经验Bayes估计优于极大似然估计,并且在考虑环境因子的情形下,Rayleigh部件可靠性指标的估计优于未考虑环境因子时的估计。     

Nonparametric estimation for accelerated life tests under intermittent inspection

This paper considers nonparametric estimation of lifetime distribution based on grouped data from constant stress accelerated life tests under intermittent inspection in which test items are inspected only at specified points in time. A method of estimating the lifetime distribution at use condition stress is proposed for the case where the time transformation function relating stress to lifetime is a version of inverse power law. Numerical studies show that the proposed method is comparable to the maximum likelihood method for small sample size and is more accurate than existing nonparametric methods used for continuous inspection. The method performs better than the maximum likelihood method when the underlying lifetime distribution is incorrectly specified.     

Statistical inference for constant-stress accelerated life tests with dependent competing risks from Marshall-Olkin bivariate exponential distribution

This paper considers a constant-stress accelerated dependent competing risks model under Type-II censoring. The dependent structure between competing risks is modeled by a Marshall-Olkin bivariate exponential distribution, and the accelerated model is described by the power rule model. The point and interval estimation of the model parameters and the reliability function under the normal usage condition at mission time are obtained by using the maximum likelihood estimation method and the bootstrap sampling technique. Moreover, the pivotal quantities based estimation are adopted to estimate the model parameters and the generalized confidence intervals. As a comparison, we also consider the Bayes estimation and the highest posterior density credible intervals for the model parameters based on conjugate priors and importance sampling method, respectively. To illustrate the proposed methodology, a Monte Carlo simulation is used to study the performances of different estimation methods. Finally, a dataset is analyzed for illustrative purpose and a comparison with the original results is also given.     

The beta exponential distribution

The exponential distribution is perhaps the most widely applied statistical distribution for problems in reliability. In this note, we introduce a generalization—referred to as the beta exponential distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of the beta exponential distribution. We derive expressions for the moment generating function, characteristic function, the first four moments, variance, skewness, kurtosis, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy, the distribution of sums and ratios, and the asymptotic distribution of the extreme order statistics. We also discuss simulation issues, estimation by the methods of moments and maximum likelihood and provide an expression for the Fisher information matrix. We hope that this generalization will attract wider applicability in reliability.     

聚氯乙烯薄膜使用寿命预测

文中对聚氯乙烯(PVC)薄膜在实际应用环境中的使用寿命进行了快速可靠的预测。通过加速寿命试验,以断裂伸长率下降至其初始值的50%为失效指标,对PVC薄膜在户外环境因子(紫外光和热)协同作用下的老化动力学进行了研究,利用威布尔统计方程分析老化失效数据,并使用Temperature-non-thermal模型建立外推关系式,确立了PVC薄膜的寿命-环境因子统计学关系式,模拟了PVC薄膜寿命分布在环境因子(热和强紫外光)协同作用下的变化规律。最后,应用上述寿命-环境因子关系式,对PVC薄膜在上海地区的户外使用寿命进行了预测,得出150μm厚的PVC薄膜曝露于上海地区时,户外使用寿命为13个月。     

Different Estimation Methods for Constant Stress Accelerated Life Test under the Family of the Exponentiated Distributions

The Accelerated Life Testing (ALT) has been used for a long time in several fields to obtain information on the reliability of product components and materials under operating conditions in a much shorter time. One of the main purposes of applying ALT is to estimate the failure time functions and reliability performance under normal conditions. This paper concentrates on the estimation procedures under ALT and how to select the best estimation method that gives accurate estimates for the reliability function. For this purpose, different estimation methods are used, such as maximum likelihood, least squares (LS), weighted LS, and probability weighted moment. Moreover, the reliability function under usual conditions is predicted. The estimation procedures are applied under the family of the exponentiated distributions in general, and for the exponentiated inverted Weibull (EIW) as a special case. Numerical analysis including simulated data and a real life data set is conducted to compare the performances between these four methods. It is found that the ML method gives the best results among other estimation methods. Finally, a comparison between the EIW and the Inverted Weibull (IW) distributions based on a real life data set is made using a likelihood ratio test. It is observed that the EIW distribution can provide better fitting than the IW in case of ALT. Copyright © 2015 John Wiley & Sons, Ltd.     

E-Bayesian estimation of reliability characteristics of two-parameter bathtub-shaped lifetime distribution with application

This article presents the expected Bayesian (E-Bayesian) estimation of the scale parameter, reliability and failure rate functions of two-parameter bathtub-shaped lifetime distribution under type-II censoring data with. Squared error loss function and gamma distribution as a conjugate prior distribution for the unknown parameter are used to obtain the E-Bayesian estimators. Also, three different prior distributions for the hyperparameters for the E-Bayesian estimators are considered. Some properties of the E-Bayesian estimators are studied. Using minimum mean square error criteria, a simulation study is conducted to compare the performance of the E-Bayesian estimators and the corresponding Bayes and maximum likelihood estimators. A real data set is analysed to show the applicability of the different proposed estimators. The numerical results show that the E-Bayesian estimators perform better than the classical and Bayesian estimators.     

Interval estimation of the tail index of a GARCH(1,1) model

It is known that the tail index of a GARCH model is determined by a moment equation, which involves the underlying unknown parameters of the model. A?tail index estimator can therefore be constructed by solving the sample moment equation with the unknown parameters being replaced by its quasi-maximum likelihood estimates (QMLE). To construct a confidence interval for the tail index, one needs to estimate the non-trivial asymptotic variance of the QMLE. In this paper, an empirical likelihood method is proposed for interval estimation of the tail index. One advantage of the proposed method is that interval estimation can still be achieved without having to estimate the complicated asymptotic variance. A?simulation study confirms the advantage of the proposed method.     

Time‐between‐events control charts for an exponentiated class of distributions of the renewal process

Time‐between‐events control charts are commonly used to monitor high‐quality processes and have several advantages over the ordinary control charts. In this article, we present some new control charts based on the renewal process, where a class of absolutely continuous exponentiated distributions is assumed for the time between events. This class includes the generalized exponential, generalized Rayleigh, and exponentiated Pareto distributions. Although we discuss the design structure for all the mentioned distributions, our main focus will be on the generalized exponential distribution due to its practical relevance and popularity. Since the generalized exponential distribution is a generalization of the traditional exponential distribution, the new control chart is more flexible than the existing exponential time‐between‐events charts. The control chart performance is evaluated in terms of some useful measures, including the average run length (ARL), the expected quadratic loss, continuous ranked probability, and the relative ARL. The effect of parameter estimation using the maximum likelihood and Bayesian methods on the ARL is also discussed in this article. The study also presents an illustrative example and 4 case studies to highlight the practical relevance of the proposal.     

A two-parameter parsimonious bathtub model for the analysis of system failure times with illustrations to reliability data

In this study, a two-parameter, upper-bounded probability distribution called the tau distribution is introduced and its applications in reliability engineering are presented. Each of the parameters of the tau distribution has a clear semantic meaning. Namely, one of them determines the upper bound of the distribution, while the value of the other parameter influences the shape of the cumulative distribution function. A remarkable property of this new probability distribution is that its probability density function, survival function, hazard rate function (HRF), and quantile function can all be expressed in terms of its cumulative distribution function. The HRF of the proposed probability distribution can exhibit an increasing trend and various bathtub shapes with or without a low and long-flat phase (useful time phase), which makes this new distribution suitable for modeling a wide range of real-world problems. The constraint maximum likelihood estimation, percentile estimation, approximate Bayesian computation, and approximate quantile estimation computation are proposed to calculate the unknown parameters of the model. The suitability of the estimation methods is verified with the aid of simulation and real-world data results. The modeling capability of the tau distribution was compared with that of some well-known two- and three-parameter probability distributions using two data sets known from the literature of reliability engineering: time between failures data of a machining center, and time to failure of data acquisition system cards. Based on empirical results, the new distribution may be viewed as a viable competitor to the Weibull, Gamma, Chen, and modified Weibull distributions.