威布尔分布下平均寿命置信限的评估方法研究

设备的平均寿命是可靠性研究中的的一个重要指标.对威布尔分布来说,由于平均寿命没有明显的枢轴量,因此给出平均寿命的精确的置信限较为困难.本文分别利用广义枢轴量、WCF展开以及三阶法三种方法,得到了设备寿命服从威布尔分布时的平均寿命的(近似)置信下限.最后对上述三种方法分别进行了模拟比较,结果显示文中给出的方法对于中小样本情形下得到的平均寿命的置信限是比较精确的.     

基于混合Ⅰ型删失数据的威布尔模型精确推断（英文）

威布尔分布是可靠性和寿命测试试验中常用的模型.本文中,我们考虑了基于混合Ⅰ型删失数据的威布尔模型精确推断.我们得到了威布尔分布未知参数最大似然估计的精确分布以及基于精确分布的置信区间.由于精确分布函数较为复杂,我们也给出了未知参数的另外几种置信区间,比如,基于近似方法的置信区间,Bootstrap置信区间.为了评价本文的方法,我们给出了一些数值模拟的结果.     

极值分布和威布尔分布异常数据的检验方法

本文对威布尔分布的极值分布异常数据的检验给出了一系列的方法，首先，导入了极值分布下一般Ｄｉｘｏｎ型统计量的精确分布，同时还给出了改进的Ｇ型统计量，及它们的分位点表。最后本文提出了一个新的统计量；Ｆ型统计量，并用Ｍｏｎｔｅ－Ｃａｒｌｏ模拟的方法给出其分位点表，从而首次给出威布尔分布异常值的直接检验方法。本文进一步讨论了这些检验方法的功效，且表明Ｆ型检验是最优的。     

一类可修威布尔型设备可用度的Fiducial推断

本文对于可维修的威布尔型设备考虑一类修如新模型,导出了在该模型下设备在任意时刻的可用度函数;基于设备寿命试验的完全数据,给出了威布尔分布在任意时刻可靠度的Fiducial分布,由此进一步求出可修威布尔型设备可用度的点估计和置信下限。最后进行了模拟运算,模拟结果表明,该方法在小样本情况下能够作出较精确的推断。     

基于混合Ⅱ型删失数据的Weibull模型精确推断和可接受抽样计划(英文)

本文考虑基于混合Ⅱ型删失数据的Weibull模型精确推断和可接受抽样计划.得到威布尔分布未知参数最大似然估计的精确分布以及基于精确分布的置信区间.由于精确分布函数较为复杂,给出未知参数的另外几种置信区间,基于近似方法的置信区间.为了评价本文的方法,给出一些数值模拟的结果.且讨论了可靠性中的可接受抽样计划问题.利用参数最大似然估计的精确分布,给出一个可接受抽样计划的执行程序和数值模拟结果.     

两参数威布尔分布可靠度和可靠寿命的精确置信限

本文讨论两参数威布尔分布可靠度和可靠寿命的置信限。对于小样本的情况比较了可靠度的MLE和基于极值分布参数的BLIE,BLUE所得的可靠度的点估计和置信下限的优劣。给出了求可靠度置信下限的用表。     

基于联合II型删失数据的两总体指数模型的精确统计推断（英文）

在本文中,我们讨论两指数总体的位置参数和尺度参数的统计推断问题.利用极大似然方法,在联合II型删失数据的情形下给出参数的精确分布以及相关精确统计推断结果.将枢轴量表示为标准指数随机变量的线性函数,并且给出枢轴量的条件精确分布,这个条件精确分布的一个很大优点是计算比较简单.利用条件精确分布,可以获得枢轴量的精确分位数.为了说明本文方法的优劣,我们也提供Bootstrap方法构造参数置信区间的相关结果.最后将理论结果,进行了部分数值模拟实验,这些数值结果列在相应的表格里.     

威布尔分布下恒加试验中的区间估计

在恒定应力下加速寿命试验中，假定在各应力下产品的寿命分布为威布尔分布Ｗ（ｍ，η），对于定数截尾样本，求分布参数和加速方程中系数的线性无偏估计（ＢＬＵＥ或ＧＬＵＥ），在此基础上本文通过构造某些枢轴量，并用Ｍｏｔｅ－Ｃａｒｌｏ方法给出了正常应力下产品可靠性特征量的置信区间，最后给出了一个数值例子。     

修如新模型中备件量的计算及其置信上限的Fiducial推断

本文研究了在修如新模型下, 对预定贮存期为$T$同时开始贮存的$N$个系统, 给出在$P_0$可修复率下所需备件数的计算公式; 针对贮存寿命服从威布尔分布的系统, 利用枢轴量, 在$P_0$可修复率和预定贮存期为$T$的条件下, 给出$N$个系统所需备件数的置信上限的定义; 并基于系统寿命试验的完全样本, 利用Fiducial方法得出备件数置信上限的计算方法.     

在定数截尾样本下三参数威布尔分布的矩估计

本文讨论了在定数截尾样本下三参数威布尔分布的矩估计问题．在定数截尾情形下，将威布尔分布数据转化为均匀分布数据，利用平均剩余寿命构造样本矩，同时，第三阶矩方程用样本的第一个次序统计量来代替，得到了在定数截尾样本下三参数威布尔分布的矩估计方程，用随机模拟方法得出了矩估计的偏性和均方误差．并与近似MLE进行了比较，表明此矩估计方法有较好的性质．     

Fitting the three-parameter Weibull distribution with Cross Entropy

The Weibull distribution is widely used in applications such as reliability and lifetime studies. Although this distribution has three parameters, for simplicity, literature pertaining to Weibull parameter estimation relaxes one of its parameters in order to estimate the other two. When the three-parameter Weibull distribution is of interest, the estimation procedure is complicated. For example, the likelihood function for a three-parameter Weibull distribution is hard to maximize. In this paper, a Cross Entropy (CE) method is developed in the context of maximum likelihood estimation (MLE) of a three-parameter Weibull distribution. Performing a simulation study, a comparative analysis between the newly developed method and two existing methods is conducted. The results show the proposed method has better performance in terms of accuracy, precision and run time for different parameter settings and sample sizes.     

A note on derivation of the generating function for the right truncated Rayleigh distribution

An expression is obtained for the probability that a Weibull random variable falls after the truncation and within a finite interval. However small, the truncation in the Weibull distribution (when the value of the shape parameter is two, it is called the Rayleigh distribution) has an impact. An attempt is made to obtain generating functions for two fixed shape parameters.     

Estimating the parameters of Weibull distribution and the acceleration factor from hybrid partially accelerated life test

This article considers the estimation of parameters of Weibull distribution based on hybrid censored data. The parameters are estimated by the maximum likelihood method under step-stress partially accelerated test model. The maximum likelihood estimates (MLEs) of the unknown parameters are obtained by Newton–Raphson algorithm. Also, the approximate Fisher information matrix is obtained for constructing asymptotic confidence bounds for the model parameters. The biases and mean square errors of the maximum likelihood estimators are computed to assess their performances through a Monte Carlo simulation study.     

On the credibility of insurance claim frequency: Generalized count models and parametric estimators

We analyze the concept of credibility in claim frequency in two generalized count models–Mittag-Leffler and Weibull count models–which can handle both underdispersion and overdispersion in count data and nest the commonly used Poisson model as a special case. We find evidence, using data from a Danish insurance company, that the simple Poisson model can set the credibility weight to one even when there are only three years of individual experience data resulting from large heterogeneity among policyholders, and in doing so, it can thus break down the credibility model. The generalized count models, on the other hand, allow the weight to adjust according to the number of years of experience available. We propose parametric estimators for the structural parameters in the credibility formula using the mean and variance of the assumed distributions and a maximum likelihood estimation over a collective data. As an example, we show that the proposed parameters from Mittag-Leffler provide weights that are consistent with the idea of credibility. A simulation study is carried out investigating the stability of the maximum likelihood estimates from the Weibull count model. Finally, we extend the analyses to multidimensional lines and explain how our approach can be used in selecting profitable customers in cross-selling; customers can now be selected by estimating a function of their unknown risk profiles, which is the mean of the assumed distribution on their number of claims.     

Reliability tests for Weibull distribution with variational shape parameter based on sudden death lifetime data

In the traditional design of reliability tests for assuring the mean time to failure (MTTF) in Weibull distribution with shape and scale parameters, it has been assumed that the shape parameter in the acceptable and rejectable populations is the same fixed number. For the purpose of expanding applicability of the reliability testing, Hisada and Arizono have developed a reliability sampling scheme for assuring MTTF in the Weibull distribution under the conditions that shape parameters in the both populations do not necessarily coincide, and are specified as interval values, respectively. Then, their reliability test is designed using the complete lifetime data. In general, the reliability testing based on the complete lifetime data requires the long testing time. As a consequence, the testing cost becomes sometimes expensive. In this paper, for the purpose of an economical plan of the reliability test, we consider the sudden death procedure for assuring MTTF in Weibull distribution with variational shape parameter.     

On nonlinear weighted errors-in-variables parameter estimation problem in the three-parameter Weibull model

This paper is concerned with the three-parameter Weibull distribution which is widely used as a model in reliability and lifetime studies. In practice, the Weibull model parameters are not known in advance and must be estimated from a random sample. Difficulties in applying the method of maximum likelihood to three-parameter Weibull models have led to a variety of alternative approaches in the literature. In this paper we consider the nonlinear weighted errors-in-variables (EIV) fitting approach. As a main result, two theorems on the existence of the EIV estimate are obtained. An illustrative example is also included.     

Modeling heterogeneity for bivariate survival data by the compound Poisson distribution with random scale

We propose a bivariate Weibull regression model with heterogeneity (frailty or random effect) which is generated by compound Poisson distribution with random scale. We assume that the bivariate survival data follow bivariate Weibull of Hanagal (2004). There are some interesting situations like survival times in genetic epidemiology, dental implants of patients and twin births (both monozygotic and dizygotic) where genetic behavior (which is unknown and random) of patients follows a known frailty distribution. These are the situations which motivate us to study this particular model. We propose a two stage maximum likelihood estimation procedure for the parameters in the proposed model and develop large sample tests for testing significance of regression parameters.     

左截尾双参数指数分布的可靠寿命的广义置信下限

本文基于左截尾双参数指数分布定数截尾数据,利用Weerahandi给出的广义枢轴量和广义置信区间的概念,通过两种不同的方法建立了可靠寿命的广义置信下限.第1种方法利用位置参数无限制时可靠寿命的广义置信下限来定义左截尾情形下可靠寿命的限制广义置信下限,第2种方法基于广义枢轴量在限制参数空间上的条件分布给出可靠寿命的条件广义置信下限.我们分别研究了这两种置信下限的性质,给出了简单易行的数值计算方法.模拟比较表明限制广义置信下限具有好的覆盖率性质,条件广义置信下限的覆盖率与参数取值有关,但它有时比限制广义置信下限具有更大均值和更小标准差.     

Empirical Bayes inference for the Weibull model

In this study, the theory of statistical kernel density estimation has been applied for deriving non-parametric kernel prior to the empirical Bayes which frees the Bayesian inference from subjectivity that has worried some statisticians. For comparing the empirical Bayes based on the kernel prior with the fully Bayes based on the informative prior, the mean square error and the mean percentage error for the Weibull model parameters are studied based on these approaches under both symmetric and asymmetric loss functions, via Monte Carlo simulations. The results are quite favorable to the empirical Bayes that provides better estimates and outperforms the fully Bayes for different sample sizes and several values of the true parameters. Finally, a numerical example is given to demonstrate the efficiency of the empirical Bayes.     

Noninformative priors for the ratio of the shape parameters of two Weibull distributions

The Weibull distribution is one of the most widely used lifetime distributions in reliability engineering. Here, the noninformative priors for the ratio of the shape parameters of two Weibull models are introduced. The first criterion used is the asymptotic matching of the coverage probabilities of Bayesian credible intervals with the corresponding frequentist coverage probabilities. We develop the probability matching priors for the ratio of the shape parameters using the following matching criteria: quantile matching, matching of the distribution function, highest posterior density matching, and matching via inversion of the test statistics. We obtain one particular prior that meets all the matching criteria. Next, we derive the reference priors for different groups of ordering. Our findings show that some of the reference priors satisfy a first-order matching criterion and the one-at-a-time reference prior is a second-order matching prior. Lastly, we perform a simulation study and provide a real-world example.