Stochastic EM algorithms for parametric and semiparametric mixture models for right-censored lifetime data

Mixture models in reliability bring a useful compromise between parametric and nonparametric models, when several failure modes are suspected. The classical methods for estimation in mixture models rarely handle the additional difficulty coming from the fact that lifetime data are often censored, in a deterministic or random way. We present in this paper several iterative methods based on EM and Stochastic EM methodologies, that allow us to estimate parametric or semiparametric mixture models for randomly right censored lifetime data, provided they are identifiable. We consider different levels of completion for the (incomplete) observed data, and provide genuine or EM-like algorithms for several situations. In particular, we show that simulating the missing data coming from the mixture allows to plug a standard R package for survival data analysis in an EM algorithm’s M-step. Moreover, in censored semiparametric situations, a stochastic step is the only practical solution allowing computation of nonparametric estimates of the unknown survival function. The effectiveness of the new proposed algorithms are demonstrated in simulation studies and an actual dataset example from aeronautic industry.     

Empirical likelihood inference for censored median regression model via nonparametric kernel estimation

An alternative to the accelerated failure time model is to regress the median of the failure time on the covariates. In the recent years, censored median regression models have been shown to be useful for analyzing a variety of censored survival data with the robustness property. Based on missing information principle, a semiparametric inference procedure for regression parameter has been developed when censoring variable depends on continuous covariate. In order to improve the low coverage accuracy of such procedure, we apply an empirical likelihood ratio method (EL) to the model and derive the limiting distributions of the estimated and adjusted empirical likelihood ratios for the vector of regression parameter. Two kinds of EL confidence regions for the unknown vector of regression parameters are obtained accordingly. We conduct an extensive simulation study to compare the performance of the proposed methods with that normal approximation based method. The simulation results suggest that the EL methods outperform the normal approximation based method in terms of coverage probability. Finally, we make some discussions about our methods.     

Empirical likelihood analysis of the Buckley-James estimator

The censored linear regression model, also referred to as the accelerated failure time (AFT) model when the logarithm of the survival time is used as the response variable, is widely seen as an alternative to the popular Cox model when the assumption of proportional hazards is questionable. Buckley and James [Linear regression with censored data, Biometrika 66 (1979) 429-436] extended the least squares estimator to the semiparametric censored linear regression model in which the error distribution is completely unspecified. The Buckley-James estimator performs well in many simulation studies and examples. The direct interpretation of the AFT model is also more attractive than the Cox model, as Cox has pointed out, in practical situations. However, the application of the Buckley-James estimation was limited in practice mainly due to its illusive variance. In this paper, we use the empirical likelihood method to derive a new test and confidence interval based on the Buckley-James estimator of the regression coefficient. A standard chi-square distribution is used to calculate the P-value and the confidence interval. The proposed empirical likelihood method does not involve variance estimation. It also shows much better small sample performance than some existing methods in our simulation studies.     

Change-point estimation for censored regression model

In this paper, we consider the change-point estimation in the censored regression model assuming that there exists one change point. A nonparametric estimate of the change-point is proposed and is shown to be strongly consistent. Furthermore, its convergence rate is also obtained.     

多台系统可靠性增长试验方法研究

提出了一种解决多台系统同步投试、同步停止试验和同步改进问题的新模型——指数模型．该模型充分考虑了增长过程中的各种可得信息,包括各改进阶段的失效数、未失效数和失效时间等数据．如果多台系统经过多次同步改进,并且单台系统的可靠性增长符合AMSAA模型,就可以合理地认为在每两个相邻的改进时刻之间,每台系统的失效时间服从指数分布．采用非参数方法得到多台系统在各同步停止试验时刻的可靠度,并利用最小二乘法拟合求得该模型中参数a和b的点估计值,以及参数b的置信限．通过在工程实例中对所提模型和几种已有模型计算结果的比较,说明了所提模型在解决多台系统同步可靠性增长问题中的合理性．     

基于聚类的轮廓数据质量监控方法研究

轮廓线的变点识别是质量管理的研究热点之一,当前研究多以轮廓整体变化为识别对象,而对局部变化问题研究相对较少,且更少有在发现变异时间的同时能够寻找到变化区域在个体轮廓曲线上位置的系统方法。本文针对轮廓线局部变化识别问题,提出基于小波变换和聚类分析的方法。通过仿真性能评价,并与现有方法进行比较,结果显示本方法能够在更小的差异度检测出变化并准确定位变化区域。在文章的末尾,本文采用了一个实例对该方法的效果进行验证。     

Exploring the latent segmentation space for the assessment of multiple change-point models

This paper addresses the retrospective or off-line multiple change-point detection problem. Multiple change-point models are here viewed as latent structure models and the focus is on inference concerning the latent segmentation space. Methods for exploring the space of possible segmentations of a sequence for a fixed number of change points may be divided into two categories: (i) enumeration of segmentations, (ii) summary of the possible segmentations in change-point or segment profiles. Concerning the first category, a dynamic programming algorithm for computing the top $N$ most probable segmentations is derived. Concerning the second category, a forward-backward dynamic programming algorithm and a smoothing-type forward-backward algorithm for computing two types of change-point and segment profiles are derived. The proposed methods are mainly useful for exploring the segmentation space for successive numbers of change points and provide a set of assessment tools for multiple change-point models that can be applied both in a non-Bayesian and a Bayesian framework. We show using examples that the proposed methods may help to compare alternative multiple change-point models (e.g. Gaussian model with piecewise constant variances or global variance), predict supplementary change points, highlight overestimation of the number of change points and summarize the uncertainty concerning the position of change points.     

A novel means of estimating quantiles for 2-parameter Weibull distribution under the right random censoring model

Censoring models are frequently used in reliability analysis to reduce experimental time. The three types of censoring models are type-I, type-II and random censoring. In this study, we focus on the right random censoring model. In this model, if the failure time exceeds its associated censoring time, then the failure time becomes a censored observation. In this case, many authors (see: Lee, Statistical Methods for Survival Data Analysis, 2nd Edition, Wiley, New York, 1992; Lawless, Statistical Models and Methods for Lifetime Data, Wiley, New York, 1982; Miller, Survival Analysis, Wiley, New York, 1981, among others) considered using the observed censoring time to impute the censored observation which, however, underestimates the true failure time. Herein, two methods to impute the censored observations are proposed in a right random censoring model for a 2-parameter Weibull distribution. By a Monte Carlo simulation, the quantile estimates are calculated to assess the performance of the proposed imputation methods with respect to their relative mean square error. Simulation results indicate that the two imputation methods proposed herein are superior to the method proposed by the above authors if the shape parameter of Weibull distribution exceeds 1, except for the lower quantiles.     

基于截断δ-冲击模型的不完全维修更换策略

本文在截断δ-冲击模型的基础上,考虑了因系统劣化而导致的冲击失效门限值与维修时间的变化,扩充失效状态,从而提出一种新的截断δ-冲击模型。以最小费用为目标,稳态可用度为约束条件,建立N型更换策略的不完全维修更换策略模型,并给出三种常用冲击到达间隔分布的期望寿命。最后通过算例验证模型的有效性,并对参数进行灵敏度分析。     

Reliability analysis for Weibull distribution with homogeneous heavily censored data based on Bayesian and least-squares methods

The reliability for Weibull distribution with homogeneous heavily censored data is analyzed in this study. The universal model of heavily censored data and existing methods, including maximum likelihood, least-squares, E-Bayesian estimation, and hierarchical Bayesian methods, are introduced. An improved method is proposed based on Bayesian inference and least-squares method. In this method, the Bayes estimations of failure probabilities are focused on for all the samples. The conjugate prior distribution of failure probability is set, and an optimization model is developed by maximizing the information entropy of prior distribution to determine the hyper-parameters. By integrating the likelihood function, the posterior distribution of failure probability is then derived to yield the Bayes estimation of failure probability. The estimations of reliability parameters are obtained by fitting distribution curve using least-squares method. The four existing methods are compared with the proposed method in terms of applicability, precision, efficiency, robustness, and simplicity. Specifically, the closed form expressions concerning E-Bayesian estimation and hierarchical Bayesian methods are derived and used. The comparisons demonstrate that the improved method is superior. Finally, three illustrative examples are presented to show the application of the proposed method.     

A sequential multiple change-point detection procedure via VIF regression

In this paper, we propose a procedure for detecting multiple change-points in a mean-shift model, where the number of change-points is allowed to increase with the sample size. A theoretic justification for our new method is also given. We first convert the change-point problem into a variable selection problem by partitioning the data sequence into several segments. Then, we apply a modified variance inflation factor regression algorithm to each segment in sequential order. When a segment that is suspected of containing a change-point is found, we use a weighted cumulative sum to test if there is indeed a change-point in this segment. The proposed procedure is implemented in an algorithm which, compared to two popular methods via simulation studies, demonstrates satisfactory performance in terms of accuracy, stability and computation time. Finally, we apply our new algorithm to analyze two real data examples.     

Sequential change-point detection for mixing random sequences under composite hypotheses

The problem of sequential detection of a change-point in the density function of one-dimensional distribution of observations from a mixing random sequence is considered when both before and after a change-point this density function belongs to a certain family of distributions, i.e. in the situation of composite hypotheses. A new quality criterion for change-point detection is proposed. The asymptotic a priori lower bound for this criterion is proved for wide class of methods of change-point detection. An asymptotically optimal method of change-point detection is proposed for which this lower bound is attained asymptotically. In particular, for the case of a simple hypothesis before a change-point, this method coincides with the generalized cumulative sums (CUSUM) method.      

Hybrid control variates-based simulation method for structural reliability analysis of some problems with low failure probability

The safety analysis of systems with nonlinear performance function and small probability of failure is a challenge in the field of reliability analysis. In this study, an efficient approach is presented for approximating small failure probabilities. To meet this aim, by introducing Probability Density Function (PDF) control variates, the original failure probability integral was reformulated based on the Control Variates Technique (CVT). Accordingly, using the adaptive cooperation of the subset simulation (SubSim) and the CVT, a new formulation was offered for the approximation of small failure probabilities. The proposed formulation involves a probability term (resulting from a fast-moving SubSim) and an adaptive weighting term that refines the obtained probability. Several numerical and engineering problems, involving nonlinear performance functions and system-level reliability problems, are solved by the proposed approach and common reliability methods. Results showed that the proposed simulation approach is not only more efficient, but is also robust than common reliability methods. It also presents a good potential for application in engineering reliability problems.     

Regression Analysis of Doubly Censored Data with a Cured Subgroup under a Class of Promotion Time Cure Models

In some situations, the failure time of interest is defined as the gap time between two related events and the observations on both event times can suffer either right or interval censoring.Such data are usually referred to as doubly censored data and frequently encountered in many clinical and observational studies. Additionally, there may also exist a cured subgroup in the whole population,which means that not every individual under study will experience the failure time of interest eventually. In this paper, we consider regression analysis of doubly censored data with a cured subgroup under a wide class of flexible transformation cure models. Specifically, we consider marginal likelihood estimation and develop a two-step approach by combining the multiple imputation and a new expectation-maximization(EM) algorithm for its implementation. The resulting estimators are shown to be consistent and asymptotically normal. The finite sample performance of the proposed method is investigated through simulation studies. The proposed method is also applied to a real dataset arising from an AIDS cohort study for illustration.     

定数双截尾样本下k/N(G)系统可靠性指标的经验Bayes估计

在定数双截尾样本下,研究了不可修k/N(G)系统可靠性指标的估计问题.将Bayes方法和极大似然法相结合获得了部件失效率、系统可靠度和平均寿命的经验Bayes估计.最后给出随机模拟例子,说明了本文方法的正确性.     

Weighted Empirical Likelihood Ratio Confidence Intervals for the Mean with Censored Data

We propose a procedure to construct the empirical likelihood ratio confidence interval for the mean using a resampling method. This approach leads to the definition of a likelihood function for censored data, called weighted empirical likelihood function. With the second order expansion of the log likelihood ratio, a weighted empirical likelihood ratio confidence interval for the mean is proposed and shown by simulation studies to have comparable coverage accuracy to alternative methods, including the nonparametric bootstrap-t. The procedures proposed here apply in a unified way to different types of censored data, such as right censored data, doubly censored data and interval censored data, and computationally more efficient than the bootstrap-t method. An example of a set of doubly censored breast cancer data is presented with the application of our methods.     

Software reliability assessment models based on cumulative bernoulli trial processes

The binomial software reliability growth model (SRGM) contains most existing SRGMs proposed in earlier work as special cases, and can describe every software failure-occurrence pattern in continuous time. In this paper, we propose generalized binomial SRGMs in both continuous and discrete time, based on the idea of cumulative Bernoulli trials. It is shown that the proposed models give some new unusual discrete models as well as the well-known continuous SRGMs. Through numerical examples with actual software failure data, two estimation methods for model parameters with grouped data are provided, and the predictive model performance is examined quantitatively.     

基于性能退化的动量轮可靠性建模与评估

为解决长寿命卫星动量轮可靠性评估无失效数据问题,研究基于性能退化的动量轮可靠性建模与评估方法.首先通过失效分析,提出以温度作为动量轮可靠性的特征量.然后利用动量轮温度的遥测数据,建立其温度随时间变化的叠合模型.为解决动量轮试验样本量不足问题,采用Bootstrap仿真方法建立动量轮寿命分布模型,评估其可靠性水平.实例表明,方法能够在无失效数据情况下,利用性能退化数据评估动量轮可靠性,且具有较高精度,为解决无失效数据条件下长寿命卫星动量轮可靠性建模与评估问题提供了一条有效途径.     

The reliability analysis of probabilistic and interval hybrid structural system

The aim of this paper is to evaluate the reliability of probabilistic and interval hybrid structural system. The hybrid structural system includes two kinds of uncertain parameters—probabilistic parameters and interval parameters. Based on the interval reliability model and probabilistic operation, a new probabilistic and interval hybrid reliability model is proposed. Firstly, we use the interval reliability model to analyze the performance function, and then sum up reliability of all regions divided by the failure plane. Based on the presented optimal criterion enumerating the main failure modes of hybrid structural system and the relationship of failure modes, the reliability of structure system can be obtained. By means of the numerical examples, the hybrid reliability model and the traditional probabilistic reliability model are critically contrasted. The results indicate the presented reliability model is more suitable for analysis and design of these structural systems and it can ensure the security of system well, and it only needs less uncertain information.     

基于广义逆理论的步进应力加速寿命试验的最优设计

步进应力加速寿命试验的最优设计问题早在20世纪60年代就已提出.本文在应力个数和未知参数个数不相等的情况下,以MLE渐近方差最小为准则,运用广义加号逆的理论,解决指数分布时定数截尾寿命试验和定时截尾寿命试验的最优设计问题.通过理论推导,得出定数截尾寿命试验时的不同应力下失效产品数的安排,以及定时截尾寿命试验时的最优应力变换点.并且,对每一种情况进行了数据模拟,结果显示,这些理论和方法在实际运用中是可行的和有效的.