Estimation of the Parameters of a Mixture of Exponential and Weibull Distributions with Progressive Censoring

A method is proposed for estimating the parameters of a mixture of exponential and Weibull distributions using censored samples. Preliminary estimates obtained by graphical analysis are refined by the method of maximum likelihood. The efficiency of the method is confirmed by the results of a statistical modeling.     

A fatigue model with local sensitivity analysis

The goal of this paper is two fold. First, it introduces a general parametric lifetime model for high‐cycle fatigue regime derived from physical, statistical, engineering and dimensional analysis considerations. The proposed model has two threshold parameters and three Weibull distribution parameters. A two‐step procedure is presented to estimate the parameters. In the first step, the two threshold parameters are estimated by minimizing a least squares regression function. In the second step, the parameters are estimated by the maximum likelihood method after pooling together the data from different stress levels. Since parameter estimation should always be accompanied by a sensitivity analysis of the fitted model, the second goal of this paper is to propose a method for sensitivity analysis for fatigue models. We show that the proposed sensitivity analysis methods are general and can be applied to any fatigue or lifetime model, not just to the one proposed here. Although several fatigue models have been proposed in the literature, to our knowledge this is the first attempt to produce methods for sensitivity analysis for fatigue models. The proposed method makes use of the well‐known duality property of mathematical programming, which states that the partial derivatives of the primal objective function with respect to the constraints right hand side parameters are the optimal values of the negative of the dual problem variables. For the parameters or data, for which sensitivities are sought, to appear on the right hand side, they are converted into artificial variables and set to their actual values, thus obtaining the desired constraints. Both the estimation and sensitivity analysis methods are illustrated by two examples, one application using real fatigue data and the other using simulated data. In addition, the sensitivity proposed method is also applied to an alternative fatigue model. Finally, some specific conclusions and recommendations are also given.     

威布尔分布的环保型电子节能灯寿命的极大似然估计

提出了一种利用改进的极大似然估计法对基于威布尔分布的环保型电子节能灯寿命数据进行分析的方法.该方法利用加速寿命实验获取环保型电子节能灯使用寿命的数据,利用统计学的方法和威布尔分布模型,实现高应力下的实验时间的等效折算.采用改进的极大似然估计,有益于对环保型电子节能灯的寿命数据进行分析.     

COMPARISONS OF OPTIMAL ACCELERATED TEST PLANS FOR ESTIMATING QUANTILES OF LIFETIME DISTRIBUTION AT THE USE CONDITION

Accelerated life tests (ALTs) and accelerated degradation tests (ADTs) are widely used for the reliability assessment of components or materials. In an ALT, failure times of test units are observed while in an ADT the failure-causing performance characteristic is measured. This article develops optimal ALT and ADT plans for estimating the qth Quantile of the lifetime distribution at the use condition, the latter being an extension of Park and Yum. Then, the two test plans are compared in terms of the asymptotic efficiency in estimating the qth quantile and of the robustness to the mis-specifications of failure probabilities. Computational results show that the ADT provides a more precise estimator than the corresponding ALT, especially when the failure probabilities are small. Concerning the robustness of a test plan to the departures of the guessed failure probabilities from their true values, neither plan completely dominates the other.     

Nonparametric estimation of the offspring distribution and the mean for a controlled branching process

In this paper, under a nonparametric context, some estimators for the mean and the offspring distribution of a controlled branching process are provided. Their conditional and unconditional moments and their asymptotic properties (consistency, normality) are investigated. Finally, as an illustration, a simulated example is presented. Research supported by the Plan Nacional de Investigación Científica, Desarrollo e Innovación Tecnológica, grant BFM2000-0356 and the Consejería de Educación, Ciencia y Tecnología de la Junta de Extremaduray y el Fondo Social Europeo, grant IPR00A056.     

Estimation of Parameters of Universal Distribution of Moments of Failure of Products

A method is proposed for estimating the parameters of a mixture of exponential and Weibull distributions for which the accuracy of preliminary estimates obtained by graphical analysis is refined in accordance with the criterion of maximum likelihood. The efficiency of the proposed method is supported by the results of statistical modeling.     

The Constant Shape Parameter Assumption in Weibull Regression

ABSTRACT

The usual assumption in Weibull regression is that the scale parameter is a function of the predictor variables, and the shape parameter is constant. We consider the problem of estimating parameters in the presence of a nonconstant shape parameter and the effect of assuming a constant shape parameter when it really is not constant. We consider both classical and Bayesian methods of estimation. The misspecification of a constant shape parameter can lead to a loss of power for tests regarding the slope parameters. We find that prediction intervals can be inaccurate when the shape parameter is incorrectly assumed to be constant.     

Weighted least‐squares estimation of the shape parameter of the Weibull distribution

The purpose of this paper is to propose the weighted least‐squares procedure for estimating the shape parameter of the Weibull distribution. Results from simulation studies illustrate the mean‐squared error of the weighted least‐squares estimator is smaller than competing procedures in all cases considered. Copyright © 2001 John Wiley & Sons, Ltd.