Parameter estimations for generalized exponential distribution under progressive type-I interval censoring

The estimates, via maximum likelihood, moment method and probability plot, of the parameters in the generalized exponential distribution under progressive type-I interval censoring are studied. A simulation is conducted to compare these estimates in terms of mean squared errors and biases. Finally, these estimate methods are applied to a real data set based on patients with plasma cell myeloma in order to demonstrate the applicabilities.     

Estimating the width of a uniform distribution when data are measured with additive normal errors with known variance

The problem of estimating the width of the symmetric uniform distribution on the line when data are measured with normal additive error is considered. The main purpose is to discuss the efficiency of the maximum likelihood estimator and the moment method estimator. It is shown that the model is regular and that the maximum likelihood estimator is more efficient than the moment method estimator. A sufficient condition is also given for the existence of both estimators.     

双边定数截尾下广义指数分布的贝叶斯估计

在双边定数截尾样本下,给出了两参数广义指数分布参数的贝叶斯估计。基于共轭先验分布,分别在刻度平方误差损失函数和Linex损失函数下,给出参数的贝叶斯估计和多层贝叶斯估计,运用随机模拟方法对各种估计结果的优良性进行了分析比较。     

后向散射系数的贝叶斯估计

后向散射系数是合成孔径雷达图像中重要的物理参数.由于合成孔径雷达测量系统的噪声干扰和其他不确定因素影响使得测量数据往往不够精确,这就需要对测量数据进行合理估计.为了对后向散射系数做出准确合理的估计,文章将后向散射系数的先验知识考虑进去,给出了后向散射系数的三种贝叶斯估计算法.贝叶斯估计的关键是概率密度模型的选取.例中选用贝塔(Beta)分布作为先验概率密度模型,伽玛(Gamma)分布作为条件概率密度模型得到了合理的估计结果,并与最大似然估计(ML)算法进行了比较,比较结果表明在对后向散射系数的估计中,贝叶斯估计算法要明显优于最大似然估计算法.     

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.     

Fuzzy Bayesian system reliability assessment based on prior two‐parameter exponential distribution under different loss functions

The fuzzy Bayesian system reliability assessment based on prior two‐parameter exponential distribution under squared error symmetric loss function and precautionary asymmetric loss function is proposed in this paper. In order to apply the Bayesian approach, the fuzzy parameters are assumed as fuzzy random variables with fuzzy prior distributions. Because the goal of the paper is to obtain fuzzy Bayes point estimators of system reliability assessment, prior distributions of location‐scale family has been changed to scale family with change variable. On the other hand, also the computational procedures to evaluate the membership degree of any given Bayes point estimate of system reliability have been provided. In order to achieve this purpose, we transform the original problem into a non‐linear programming problem. This non‐linear programming problem is then divided into four sub‐problems for the purpose of simplifying computation. Finally, the sub‐problems can be solved by using any commercial optimizers, e.g. GAMS or LINGO. Copyright © 2010 John Wiley & Sons, Ltd.     

Empirical Bayes estimation of the scale parameter in a Pareto distribution

Let f(xθ) = αθ^αx^−(α+1)I(x>θ) be the pdf of a Pareto distribution with known shape parameter α>0, and unknown scale parameter θ. Let {(X_i, θ_i)} be a sequence of independent random pairs, where X_i's are independent with pdf f(xα_i), and θ_i are iid according to an unknown distribution G in a class of distributions whose supports are included in an interval (0, m), where m is a positive finite number. Under some assumption on the class and squared error loss, at (n + 1)th stage we construct a sequence of empirical Bayes estimators of θ_n+1 based on the past n independent observations X₁,…, X_n and the present observation X_n+1. This empirical Bayes estimator is shown to be asymptotically optimal with rate of convergence O(n^−1/2). It is also exhibited that this convergence rate cannot be improved beyond n^−1/2 for the priors in class .     

GFREG: a computer program for maximum likelihood regression using the Generalized F distribution

A FORTRAN program is described for maximum likelihood estimation within the Generalized F family of distributions. It can be used to estimate regression parameters in a log-linear model for censored survival times with covariates, for which the error distribution may have a great variety of shapes, including most distributions of current use in biostatistics. The optimization is performed by an algorithm based on the generalized reduced gradient method. A stepwise variable search algorithm for covariate selection is included in the program. Output features include: model selection criteria, standard errors of parameter estimates, quantile and survival rates with their standard errors, residuals and several plots. An example based on data from Princess Margaret Hospital, Toronto, is discussed to illustrate the program's capabilities.     

Inference for a simple step-stress model with competing risks for failure from the exponential distribution under time constraint

In reliability analysis, accelerated life-testing allows for gradual increment of stress levels on test units during an experiment. In a special class of accelerated life tests known as step-stress tests, the stress levels increase discretely at pre-fixed time points, and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. In this article, we consider the simple step-stress model under time constraint when the lifetime distributions of the different risk factors are independently exponentially distributed. Under this setup, we derive the maximum likelihood estimators (MLEs) of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. Since it is found that the MLEs do not exist when there is no failure by any particular risk factor within the specified time frame, the exact sampling distributions of the MLEs are derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions, the parametric bootstrap method, and the Bayesian posterior distribution, we discuss the construction of confidence intervals and credible intervals for the parameters. Their performance is assessed through Monte Carlo simulations and finally, we illustrate the methods of inference discussed here with an example.     

Generalized weighted likelihood density estimators with application to finite mixture of exponential family distributions

The family of weighted likelihood estimators largely overlaps with minimum divergence estimators. They are robust to data contaminations compared to MLE. We define the class of generalized weighted likelihood estimators (GWLE), provide its influence function and discuss the efficiency requirements. We introduce a new truncated cubic-inverse weight, which is both first and second order efficient and more robust than previously reported weights. We also discuss new ways of selecting the smoothing bandwidth and weighted starting values for the iterative algorithm. The advantage of the truncated cubic-inverse weight is illustrated in a simulation study of three-component normal mixtures model with large overlaps and heavy contaminations. A real data example is also provided.