Estimation for the three-parameter lognormal distribution based on progressively censored data

Some work has been done in the past on the estimation of parameters of the three-parameter lognormal distribution based on complete and censored samples. In this article, we develop inferential methods based on progressively Type-II censored samples from a three-parameter lognormal distribution. In particular, we use the EM algorithm as well as some other numerical methods to determine maximum likelihood estimates (MLEs) of parameters. The asymptotic variances and covariances of the MLEs from the EM algorithm are computed by using the missing information principle. An alternative estimator, which is a modification of the MLE, is also proposed. The methodology developed here is then illustrated with some numerical examples. Finally, we also discuss the interval estimation based on large-sample theory and examine the actual coverage probabilities of these confidence intervals in case of small samples by means of a Monte Carlo simulation study.     

Assessing the lifetime performance index of products from progressively type II right censored data using Burr XII model

Process capability analysis has been widely applied in the field of quality control to monitor the performance of industrial processes. In practice, lifetime performance index C_L is a popular means to assess the performance and potential of their processes, where L is the lower specification limit. Nevertheless, many processes possess a non-normal lifetime model, the assumption of normality is often erroneous. Progressively censoring scheme is quite useful in many practical situations where budget constraints are in place or there is a demand for rapid testing. The study will apply data transformation technology to constructs a maximum likelihood estimator (MLE) of C_L under the Burr XII distribution based on the progressively type II right censored sample. The MLE of C_L is then utilized to develop a new hypothesis testing procedure in the condition of known L. Finally, we give two examples to illustrate the use of the testing procedure under given significance level α.     

On progressively censored competing risks data for Weibull distributions

In survival analysis, or in reliability study, an investigator is often interested in the assessment of a specific risk in the presence of other risk factors. It is well known as the competing risks problem in statistical literature. Moreover, censoring is inevitable in any life testing or reliability study. In this paper, we consider a very general censoring scheme, namely a progressive censoring scheme. It is further assumed that the lifetime distribution of the individual causes are independent and Weibull-distributed with the same shape parameters but different scale parameters. We obtain the maximum likelihood and approximate maximum likelihood estimates of the unknown parameters. We also compute the observed Fisher information matrix using the missing information principles, and use them to compute the asymptotic confidence intervals. Monte Carlo simulations are performed to compare the performances of the different methods, and one data set is analyzed for illustrative purposes. We also discuss different optimality criteria, and selected optimal progressive censoring plans are presented.     

Prediction for Pareto distribution based on progressively Type-II censored samples

In this paper, we discuss different predictors of times to failure of units censored in multiple stages in a progressively censored sample from Pareto distribution. The best linear unbiased predictors, maximum likelihood predictors and approximate maximum likelihood predictors are considered. We also present two methods for obtaining prediction intervals for the times to failure of units. A numerical simulation study involving two data sets is presented to illustrate the methods of prediction.     

Estimation of parameters from progressively censored data using EM algorithm

EM algorithm is used to determine the maximum likelihood estimates when the data are progressively Type II censored. The method is shown to be feasible and easy to implement. The asymptotic variances and covariances of the ML estimates are computed by means of the missing information principle. The methodology is illustrated with two popular models in lifetime analysis, the lognormal and Weibull lifetime distributions.     

Parametric inference for progressive Type-I hybrid censored data on a simple step-stress accelerated life test model

This paper considers a simple step-stress accelerated life test model under progressive Type-I hybrid censoring scheme. The progressive Type-I hybrid censoring scheme and statistical method in synthetic accelerated stresses are provided so as to decrease the lifetime and reduce the test cost. An exponentially distributed life of test units and a cumulative exposure model are assumed. The maximum likelihood estimates of the model parameters are obtained using a pivotal quantity. Two useful lemmas and a theorem are given to construct the approximate confidence intervals for the model parameters. Finally, simulation results are provided to assess the method of inference developed in this article. The simulation results show that the method does improve for large sample size.     

Relations for moments of progressively Type-II censored order statistics from half-logistic distribution with applications to inference

In this paper, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a half-logistic distribution. The use of these relations in a systematic recursive manner would enable one to compute all the means, variances and covariances of progressively Type-II right censored order statistics from the half-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁,…,R_m). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan (1985). These moments are then utilized to derive best linear unbiased estimators of the scale and location-scale parameters of the half-logistic distribution. A comparison of these estimators with the maximum likelihood estimates is then made. The best linear unbiased predictors of censored failure times is then discussed briefly. Finally, two numerical examples are presented to illustrate all the inferential methods developed here.     

Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data

This paper describes the Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data. First we consider the Bayesian inference of the unknown parameter under a squared error loss function. Although we have discussed mainly the squared error loss function, any other loss function can easily be considered. A Gibbs sampling procedure is used to draw Markov Chain Monte Carlo (MCMC) samples, and they have in turn, been used to compute the Bayes estimates and also to construct the corresponding credible intervals with the help of an importance sampling technique. We have performed a simulation study in order to compare the proposed Bayes estimators with the maximum likelihood estimators. We further consider one-sample and two-sample Bayes prediction problems based on the observed sample and provide appropriate predictive intervals with a given coverage probability. A real life data set is used to illustrate the results derived. Some open problems are indicated for further research.     

逐步II型截尾下表决系统可靠性指标的估计

利用逐步II型截尾样本,研究一种表决系统可靠性指标的估计问题。将经典统计方法和Bayes方法相结合,在平方损失下,推导出部件失效率、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,说明了该方法的正确性。     

逐步Ⅱ型截尾下表决系统可靠性指标的估计

利用逐步Ⅱ型截尾样本,研究一种表决系统可靠性指标的估计问题.将经典统计方法和Bayes方法相结合,在平方损失下,推导出部件失效率、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,说明了该方法的正确性.     

Model based bootstrap methods for interval censored data

The performance of model based bootstrap methods for constructing point-wise confidence intervals around the survival function with interval censored data is investigated. It is shown that bootstrapping from the nonparametric maximum likelihood estimator of the survival function is inconsistent for the current status model. A model based smoothed bootstrap procedure is proposed and proved to be consistent. In fact, a general framework for proving the consistency of any model based bootstrap scheme in the current status model is established. In addition, simulation studies are conducted to illustrate the (in)-consistency of different bootstrap methods in mixed case interval censoring. The conclusions in the interval censoring model would extend more generally to estimators in regression models that exhibit non-standard rates of convergence.     

Estimation for mixed exponential distributions under type-II progressively hybrid censored samples

The type-II progressively hybrid censoring scheme can be deemed as a mixture of type-II progressive and hybrid censoring schemes, which has been utilized to analyze lifetime data in the literature for exponential distribution and Weibull distribution and so on, where the experiment terminates at a pre-specified time. However, little attention has been paid to parametric estimation under this censoring scheme for the mixed exponential distribution (MED) model, which is an important model in life data analysis. Based on type-II progressively hybrid censored samples, the estimation problem of the MED is addressed. The closed form of maximum likelihood estimators (MLEs) of unknown parameters using the EM algorithm are obtained. Some Monte Carlo simulations are implemented and a real data set is analyzed to illustrate the performance of the proposed method.     

一类软件可靠度新模型的统计推断

对软件可靠性的著名模型——JM模型进行了修改和研究,给出了模型中的参数的极大似然估计存在的充要条件。     

扩展关系数据模型下的Credal网络近似推理

对扩展关系数据模型作进一步延伸,构建了扩展乘积连接和扩展边缘连接操作。提出利用上述操作计算Credal网络中各结点联合概率分布和后验概率的方法,从而实现了Credal网络的近似推理,并给出了该近似推理的SQL语言实现算法,算例结果表明该方法是有效的。建立了Credal网络的关系数据结构,利用关系数据库的查询优化机制,有效规避了Credal网络推理过程中Credal集顶点组合爆炸问题,提高了Credal网络推理的效率。     

Inductive inference of regular languages based on model inference

This paper is concerned with an algorithm for identifying an unknown regular language from examples of its members and non-members. The algorithm is based on the model inference algorithm given by Shapiro. In our setting, however, a given first order language for describing a target logic program has countably many unary predicate symbols: q ₀,q ₁,q ₂…. On the other hand, the oracle which gives information about the unknown regular language to the inference algorithm has no interpretation for predicates other than the predicate q ₀. In such a setting,we cannot directly take advantage of the contradiction backtracing algorithm which is one of the most important parts for the efficiency of the model inference algorithm. In order to overcome this disadvantage, we develop a method for giving an interpretation for predicates other than the predicate q ₀ indirectly, which is based on the idea of using the oracle and a one to one mapping from a set of predicates to a set of strings. Furthermore, we propose a model inference algorithm for regular languages using the method, then argue the correctness and the time complexity of the algorithm     

Using the EM algorithm for inference in a mixture of distributions with censored but partially identifiable data

This paper describes an implementation of the EM algorithm for the statistical analysis of a finite mixture of distributions arising when data are censored but partially identifiable. We consider a scheme of type I censoring where censoring times are random. The estimation of standard errors proposed by Meng and Rubin (1991. Using EM to obtain asymptotic variance-covariance matrices: the SEM algorithm. J. Amer. Statist. Assoc. 86(416), 899-909) is also implemented in the context of the above mixture. A Bayesian method introduced in Contreras-Cristán et al. (2003. Statistical inference for mixtures of distributions for censored data with partial identification. Commun. in Statist. Theory Methods 32(4), 749-774) for the case of a constant censoring value is extended to the case of random censoring times. Comparisons with different methods are carried out both with simulated data and with the observations on failure times for communication transmitter-receivers of Mendenhall and Hader (1958. Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika 45, 504-520).     

Statistical inference for the lifetime performance index based on generalised order statistics from exponential distribution

In manufacturing industries, the lifetime of an item is usually characterised by a random variable X and considered to be satisfactory if X exceeds a given lower lifetime limit L. The probability of a satisfactory item is then η_L := P(X ≥ L), called conforming rate. In industrial companies, however, the lifetime performance index, proposed by Montgomery and denoted by C_L, is widely used as a process capability index instead of the conforming rate. Assuming a parametric model for the random variable X, we show that there is a connection between the conforming rate and the lifetime performance index. Consequently, the statistical inferences about η_L and C_L are equivalent. Hence, we restrict ourselves to statistical inference for C_L based on generalised order statistics, which contains several ordered data models such as usual order statistics, progressively Type-II censored data and records. Various point and interval estimators for the parameter C_L are obtained and optimal critical regions for the hypothesis testing problems concerning C_L are proposed. Finally, two real data-sets on the lifetimes of insulating fluid and ball bearings, due to Nelson (1982) Nelson, W. (1982), Applied Life Data Analysis, New York: John Wiley &; Sons.[Crossref] , [Google Scholar] and Caroni (2002) Caroni, C. (2002), ‘The correct “ball bearings” data’, Lifetime Data Analysis, 8, 395–399.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar], respectively, and a simulated sample are analysed.     

Distribution-free statistical inference for generalized Lorenz dominance based on grouped data

One income distribution is preferable to another under any increasing and Schur-concave (S-concave) social welfare function (SWF) if and only if the generalized Lorenz (GL) curve of the first distribution lies above that of the second. Thus, testing for GL dominance of one distribution over another is of interest. The paper focuses on inference based on grouped data and makes two contributions: (i) it gives a new formula for the asymptotic variance–covariance matrix of a vector of sample GL curve ordinates, interpreting it as a method-of-moments (MM) estimator, and (ii) it proposes a new test for multivariate inequality restrictions, of which GL dominance is a special case. For the Japanese household income data grouped into deciles, the test accepts the null hypothesis that income distribution in Japan improved from 1979 to 1994.     

LSS: an S-Plus/R program for the accelerated failure time model to right censored data based on least-squares principle

Due to lack of proper inference procedure and software, the ordinary linear regression model is seldom used in practice for the analysis of right censored data. This paper presents an S-Plus/R program that implements a recently developed inference procedure [Z. Jin, D.Y. Lin, Z. Ying, On least-squares regression with censored data, Biometrika 93 (2006) 147-161] for the accelerated failure time model based on the least-squares principle. The program is user-friendly and yields outputs similar to S-Plus/R function 1m.