Bayesian accelerated life test plans for series systems with Weibull component lifetimes

This article presents optimal Bayesian accelerated life test plans for series systems under Type-I censoring scheme. First, the component lifetimes are assumed to follow independent Weibull distributions. The scale parameters of Weibull lifetime distributions are related to the external stress variable through a general stress translation function. For a fixed number of design points, optimal Bayesian ALT plans are first obtained by solving constrained optimization problems under two different Bayesian design criteria. The global optimality of the resulting fixed-point optimal designs is then verified via the General Equivalence Theorem. This article also provides the optimized compromise ALT plans which are extremely useful in real-life applications. A detailed sensitivity analysis is then performed to find out the effect of various planning inputs on the resulting optimal Bayesian ALT plans. A simulation study is then conducted to visualize the resulting sampling variations from the optimal Bayesian ALT plans. Finally, this article considers a series system with dependent component lifetimes. Optimal ALT plans are obtained assuming a Gamma frailty model.     

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.     

Two sample Bayesian prediction intervals for order statistics based on the inverse exponential-type distributions using right censored sample

In this paper, two sample Bayesian prediction intervals for order statistics (OS) are obtained. This prediction is based on a certain class of the inverse exponential-type distributions using a right censored sample. A general class of prior density functions is used and the predictive cumulative function is obtained in the two samples case. The class of the inverse exponential-type distributions includes several important distributions such the inverse Weibull distribution, the inverse Burr distribution, the loglogistic distribution, the inverse Pareto distribution and the inverse paralogistic distribution. Special cases of the inverse Weibull model such as the inverse exponential model and the inverse Rayleigh model are considered.     

Optimal discriminant functions for normal populations

A class of discriminant rules which includes Fisher’s linear discriminant function and the likelihood ratio criterion is defined. Using asymptotic expansions of the distributions of the discriminant functions in this class, we derive a formula for cut-off points which satisfy some conditions on misclassification probabilities, and derive the optimal rules for some criteria. Some numerical experiments are carried out to examine the performance of the optimal rules for finite numbers of samples.     

A load share model for non-identical components of a k-out-of-m system

We consider a k-out-of-m load sharing system when the lifetimes of the components are not necessarily identically distributed random variables. For such systems, a model for the load sharing phenomenon through the exponentiated conditional survival functions of ordered failure times is proposed. This model is more general than the load sharing model with identically distributed component lifetimes and leads to a different family of distributions for ordered random variables. A general expression for the reliability of the system is given. The computations of the reliability for a two component parallel load sharing system corresponding to the exponential and Weibull distributions are discussed. For illustrative purpose, we discuss the inference procedures for a two component parallel load sharing system corresponding to the exponential distributions. A simulation study is carried out to assess the proposed estimation and testing procedures. The applicability of the proposed load sharing model is shown through two data sets.     

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.     

Statistical inference for the Weitzman overlapping coefficient in a family of distributions

A usual problem in applied statistics is the one related to the estimation of the common area under two distributions, which is usually estimated by means of an overlapping coefficient. Particularly, for the Weitzman overlapping coefficient, we found that it is possible to provide a general expression that facilitates making inferences on this coefficient, for many distributions. This expression depends only on two parameters which are actually functions of the parameters of the selected models, among which we can mention the exponential, Weibull, Gumbel, Fréchet and some other distributions that arise under certain transformations of an exponential random variable. The simplicity of our unifying proposal is illustrated considering three well known distributions that have been individually analyzed in statistical literature. To illustrate the performance of the likelihood confidence intervals obtained for this overlapping coefficient, under our proposal, we carried out some simulation studies that yielded adequate coverage frequencies, and just for the sake of comparison we also computed Bootstrap confidence intervals. A real data set is analyzed to exemplify our proposal.     

Bayesian accelerated life testing under competing log-location-scale family of causes of failure

This article provides Bayesian analyses of data arising from multi-stress accelerated life testing of series systems. The component log-lifetimes are assumed to independently belong to some log-concave location-scale family of distributions. The location parameters are assumed to depend on the stress variables through a linear stress translation function. Bayesian analyses and associated predictive inference of reliability characteristics at usage stresses are performed using Gibbs sampling from the joint posterior. The developed methodology is numerically illustrated by analyzing a real data set through Bayesian model averaging of the two popular cases of Weibull and log-normal, with the later getting a special focus in this article as a slightly easier example of the log-location-scale family. A detailed simulation study is also carried out to compare the performance of various Bayesian point estimators for the log-normal case.     

Comparisons of models for estimation of safe doses using measures of the heaviness of tail of a distribution

Summary Procedures to estimate a dose of which the incidence probability is very small (e.g. 10⁻⁶) have been developed to evaluate the safety of chemical compounds. To compare models for estimation of safe doses quantitatively, a measure of the heaviness of tail of a distribution and a measure of tail at the origin are introduced. These measures have a theoretical basis for the comparison of tail behavior between distributions. Using the two measures, a tail ordering is defined to present a criterion for the comparison of models and is discussed for the probit, the logit, the Weibull, the (generalized) multihit, the (generalized) multitarget and the multistage models. The multistage model is most conservative among them, while the probit model has the reverse property. The Weibull model is more conservative than the logit. The multihit and multitarget models are found to be more sensitive than the Weibull and the logit. The Institute of Statistical Mathematics National Institute of Environmental Health Sciences     

Identifiability of finite mixtures using a new transform

Identifiability of finite mixtures of the following families of distributions are proved: Weibull, normal log, chi, pareto and power function.     

Extended DEA-Discriminant Analysis

A new non-parametric method is recently proposed for discriminant analysis (Sueyoshi, T., 1999. DEA-discriminant analysis in the view of goal programming. European Journal of Operational Research 115, 564–582). The new approach is referred to as “DEA-Discriminant Analysis (DEA-DA)” that is designed to identify the existence of an overlap between two groups, then determining the group membership of a newly sampled observation. A unique feature of the new technique is that it does not assume any discriminant function for group classification. As an extension of his study, this research proposes a new type of DEA-DA, or “Extended DEA-DA”, that can overcome some methodological drawbacks of its original formulation, but simultaneously maintaining its discriminant capabilities. Using a real data set regarding Japanese banks and a large simulation study, this research confirms that the Extended DEA-DA outperforms conventional linear and nonlinear discriminant analysis techniques.     

A generalized log-normal distribution and its goodness of fit to censored data

In life testing experiments, the skewed distributions like log-normal, Weibull, gamma and generalized gamma are the most suitable models for recording the failure time measurements. In this paper, a generalized version of log-normal distribution is proposed and its goodness-of-fit for a randomly censored data set representing the remission times of bladder cancer patients has been demonstrated and compared with other lifetime models considered in the literature. The P-P plots of Kaplan-Meier estimator against the survival functions of the considered models are used to show the goodness-of-fit. A simulation study is also performed to estimate the parameters in both the classical and Bayesian setups.     

Graduating sample data using generalized Weibull functions

Most activities, or events, in discrete event simulation models are probabilistic in nature and describable by density functions. Consequently, persons developing simulation models are frequently required to select and fit density functions to sample data and, correspondingly, to determine the process, or random-event, generator.

This paper seeks to establish that the graduation process involved with simulation modeling may be expedited: generalized Weibull functions numerically fitted by nonlinear least-squares or maximum-likelihood procedures are comparable in performance to the fits of more traditional functions. The use of a single function simplifies the ambiguous process of function selection, while offering, in addition, the important advantages of a closed-form inverse process generator.     

Projection-pursuit approach to robust linear discriminant analysis

Discriminant analysis plays an important role in multivariate statistics as a prediction and classification method. It has been successfully applied in many fields of work and research. As it happens with other multivariate methods, discriminant analysis is highly vulnerable to the presence of outliers that commonly occur in many real world data sets. The lack of robustness of the classical estimators on which the linear discriminant function is based is a severe disadvantage and several authors have worked to find efficient ways to prevent the damage that outliers can cause. This paper focuses on the projection-pursuit approach to discriminant analysis. The projection-pursuit estimators are described and theoretical properties are deduced and their relevance is highlighted. These include Fisher consistency, affine equivariance, partial influence functions and asymptotic distributions. Application to real data and a simulation study reveal the robustness of the projection-pursuit approach. In both analyses the data relates to a large number of variables, a situation that is becoming common when new technology is applied to data gathering.     

A stochastic expectation-maximization algorithm for the analysis of system lifetime data with known signature

Statistical estimation of the model parameters of component lifetime distribution based on system lifetime data with known system structure is discussed here. We propose the use of stochastic expectation-maximization (SEM) algorithm for obtaining the maximum likelihood estimates of model parameters based on complete and censored system lifetimes. Different ways of implementing the SEM algorithm are also studied. We have shown that the proposed methods are feasible and are easy to implement for various families of component lifetime distributions. The proposed methodologies are then illustrated with two popular lifetime models—the Weibull and Birnbaum-Saunders distributions. Monte Carlo simulation is then used to compare the performance of the proposed methods with the corresponding estimation by direct maximization. Finally, two illustrative examples are presented along with some concluding remarks.     

Method of primers in calculating the probability of destruction of a unidirectional composite material in tension

A method of primers is elaborated which allows one to calculate the distribution function of durability of a composite material in tension in the reinforcement direction. Integral and differential equations for calculating the probabilities of formation of primers and destruction of a material caused by their formation are presented. Distribution functions of material strength for the parameter of Weibull distribution of fiber strength on the interval 2.1 ≤ β _f ≤ 50.1 are calculated. From the functions, the average values and root-mean-square deviations of material strength are found. The results obtained agree well with calculations by using the structural-imitation simulation. The distribution functions of material strength with a high precision are approximated by the three-parameter Weibull distributions. The distribution parameters are approximated by the linear functions of ln (β_f). __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 823–838, November–December, 2007.     

Duration dependence models for claim counts

The aim of the paper is to introduce new claim count distributions constructed from different waiting time assumptions, such as the Exponential, Gamma and Weibull distributions. These models are then fitted to panel data with Gamma distributed random effects. The random effects allow for serial dependence and take residual heterogeneity into account. Predictive distributions are obtained with the help of Markov Chain Monte Carlo simulations. The approach is illustrated on the basis of a Belgian motor third party liability insurance portfolio observed for three years.     

Concerning the glass fiber strength distribution function

The statistical strength distribution functions of glass monofilaments of various composition are considered. On the basis of the experimental data it is shown that the fiber strength distributions can be described by a three-parameter function of the Weibull type.All-Union Scientific-Research Institute of Glass-Reinforced Plastics and Glass Fibers, Moscow Regions. Translated from Mekhanika Polimerov, No. 1, pp. 131–136, January–February, 1970.     

Mixture of Weibull distributions—parametric characterization of failure rate function

A mixture distribution involving two two-parameter Weibull distributions is characterized by five parameters. The shape of the failure rate function for the mixture depends on the parameter values and can be one of eight distinct shapes. In this paper we present a complete parametric characterization of the shapes for the failure rate function for the mixture. © 1998 John Wiley & Sons, Ltd.     

Estimation of the probabilities of misclassification for a linear discriminant function in the univariate normal case

Summary The probability of misclassification inherent in the use of a linear discriminant function is not necessarily known to the experimenter using such a function. Various estimators calculated from the sample used to generate the sample discriminant function have been proposed. The purpose of this paper is to evaluate and to compare several of these estimators by using unconditional mean square error as the criterion. Discussion is restricted to the case where each of the distributions is univariate normal with common variance. This paper is partially based on the Ph.D. thesis of N. Sedransk [7] at Iowa State University.