Bayesian inference for the parameters of Weibull distribution under progressive Type-I interval censored data with beta-binomial removals

This article considers the problem of estimating the parameters of Weibull distribution under progressive Type-I interval censoring scheme with beta-binomial removals. Classical as well as the Bayesian procedures for the estimation of unknown model parameters have been developed. The Bayes estimators are obtained under SELF and GELF using MCMC technique. The performance of the estimators, has been discussed in terms of their MSEs. Further, expression for the expected number of total failures has been obtained. A real dataset of the survival times for patients with plasma cell myeloma is used to illustrate the suitability of the proposed methodology.     

Some results on maximum likelihood estimators of parameters of exponential distribution under type I progressive censoring with changing failure rates

In this paper the study of relative bias (RB), exact variance and mean square error (MSE) of the maximum likelihood estimators of the exponential distribution under type I progressive censoring with changing failure rates is considered. A minimum mean square error (MMSE) estimator for the parameter at each stage is proposed. The numerical evalution of their relative performance is made for selected values of n and p. Further results concerning group-censoring, total expected waiting time and optimal spacings of the times of censoring are derived and results obtained by Kendell and Anderson (1971) are deduced as special cases.     

Nonparametric estimation of a survival function under progressive Type-I multistage censoring

Failure time data subject to three progressive Type-I multistage censoring schemes are studied. Product limit estimators are proposed for the estimation of the survival function. It is shown that the resulting estimators are asymptotically equivalent to the corresponding estimators where the data are subject to a random iid right censoring scheme. Many well-known results on confidence bands and tests for randomly right censored data hold for these progressive censoring schemes.     

Comparison of two weibull distributions under random censoring

The two-sample problem for comparing Weibull scale parameters is studied for randomly censored data. Three different test statistics are considered and their asymptotic properties are established under a sequence of local alternatives, It is shown that both the test statistic based on the mlefs (maximum likelihood estimators) and the likelihood ratio test are asymptotically optimum. The third statistic based only on the number of failures is not, Asymptotic relative efficiency of this statistic is obtained and its numerical values are computed for uniform and Weibull censoring, Effects of uniform random censoring on the censoring level of the experiment are illus¬trated, A direct proof for the joint asymptotic normality of the mlefs of the shape and the scale parameters is also given     

Inference for the extreme value distribution under progressive Type-II censoring

The extreme value distribution has been extensively used to model natural phenomena such as rainfall and floods, and also in modeling lifetimes and material strengths. Maximum likelihood estimation (MLE) for the parameters of the extreme value distribution leads to likelihood equations that have to be solved numerically, even when the complete sample is available. In this paper, we discuss point and interval estimation based on progressively Type-II censored samples. Through an approximation in the likelihood equations, we obtain explicit estimators which are approximations to the MLEs. Using these approximate estimators as starting values, we obtain the MLEs using an iterative method and examine numerically their bias and mean squared error. The approximate estimators compare quite favorably to the MLEs in terms of both bias and efficiency. Results of the simulation study, however, show that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are unsatisfactory for both these estimators and particularly so when the effective sample size is small. We, therefore, suggest the use of unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. The results are presented for a wide range of sample sizes and different progressive censoring schemes. We conclude with an illustrative example.     

On the distribution of a test for exponentiality based on progressively type-II right censored spacings

A test for exponentiality based on progressively Type-II right censored spacings has been proposed recently by Balakrishnan et al. (2002). They derived the asymptotic null distribution of the test statistic. In this work, we utilize the algorithm of Huffer and Lin (2001) to evaluate the exact null probabilities and the exact critical values of this test statistic.     

Robust Bayesian estimation of a bathtub-shaped distribution under progressive Type-II censoring

The bathtub-shaped failure rate function has been used for modeling the life spans of a number of electronic and mechanical products, as well as for modeling the life spans of humans, especially when some of the data are censored. This article addresses robust methods for the estimation of unknown parameters in a two-parameter distribution with a bathtub-shaped failure rate function based on progressive Type-II censored samples. Here, a class of flexible priors is considered by using the hierarchical structure of a conjugate prior distribution, and corresponding posterior distributions are obtained in a closed-form. Then, based on the square error loss function, Bayes estimators of unknown parameters are derived, which depend on hyperparameters as parameters of the conjugate prior. In order to eliminate the hyperparameters, hierarchical Bayesian estimation methods are proposed, and these proposed estimators are compared to one another based on the mean squared error, through Monte Carlo simulations for various progressively Type-II censoring schemes. Finally, a real dataset is presented for the purpose of illustration.     

Bayesian estimation and prediction for the inverse weibull distribution under general progressive censoring

Abstract

This paper deals with Bayesian estimation and prediction for the inverse Weibull distribution with shape parameter α and scale parameter λ under general progressive censoring. We prove that the posterior conditional density functions of α and λ are both log-concave based on the assumption that λ has a gamma prior distribution and α follows a prior distribution with log-concave density. Then, we present the Gibbs sampling strategy to estimate under squared-error loss any function of the unknown parameter vector (α, λ) and find credible intervals, as well as to obtain prediction intervals for future order statistics. Monte Carlo simulations are given to compare the performance of Bayesian estimators derived via Gibbs sampling with the corresponding maximum likelihood estimators, and a real data analysis is discussed in order to illustrate the proposed procedure. Finally, we extend the developed methodology to other two-parameter distributions, including the Weibull, Burr type XII, and flexible Weibull distributions, and also to general progressive hybrid censoring.     

Variable selection for semiparametric proportional hazards model under progressive Type-II censoring

Variable selection is an effective methodology for dealing with models with numerous covariates. We consider the methods of variable selection for semiparametric Cox proportional hazards model under the progressive Type-II censoring scheme. The Cox proportional hazards model is used to model the influence coefficients of the environmental covariates. By applying Breslow’s “least information” idea, we obtain a profile likelihood function to estimate the coefficients. Lasso-type penalized profile likelihood estimation as well as stepwise variable selection method are explored as means to find the important covariates. Numerical simulations are conducted and Veteran’s Administration Lung Cancer data are exploited to evaluate the performance of the proposed method.     

Interval estimation for the generalized inverted exponential distribution under progressive first failure censoring

In this paper, we consider the inferential procedures for the generalized inverted exponential distribution under progressive first failure censoring. The exact confidence interval for the scale parameter is derived. The generalized confidence intervals (GCIs) for the shape parameter and some commonly used reliability metrics such as the quantile and the reliability function are explored. Then the proposed procedure is extended to the prediction interval for the future measurement. The GCIs for the reliability of the stress-strength model are discussed under both equal scale and unequal scale scenarios. Extensive simulations are used to demonstrate the performance of the proposed GCIs and prediction interval. Finally, an example is used to illustrate the proposed methods.     

Design of accelerated life test plans under progressive Type II interval censoring with random removals

This paper investigates the design of accelerated life test (ALT) plans under progressive Type II interval censoring with random removals. Units’ lifetimes are assumed to follow a Weibull distribution, and the number of random removals at each inspection is assumed to follow a binomial distribution. The optimal ALT plans, which minimize the asymptotic variance of an estimated quantile at use condition, are determined. The expected duration of the test and the expected number of inspections on each stress level are calculated. A numerical study is conducted to investigate the properties of the derived ALT plans under different parameter values. For illustration purpose, a numerical example is also given.     

Estimation for two-parameter weibull distribution and extreme-value distribution under multiply type-II censoring

Compared to Type-II censoring, multiply Type-II censoring is a more general, yet mathematically and numerically much more complicated censoring scheme. For multiply Type II censored data from a two-parameter Weibull distribution, we propose several estimators, including MLE, approximate MLE, and estimators corresponding to the BLUE and BLIE from estimating parameters in extreme-value distribution. An approximately unbiased estimator for the shape parameter is also proposed which has the smallest MSE. Numerical examples show that this estimator is the best in terms of bias and MSE. Numerical examples also show that the approximate MLE which admits a closed form is better for estimating the scale parameter.     

Robust Bayesian analysis for exponential parameters under generalized Type-II progressive hybrid censoring

The Type-II progressive hybrid censoring scheme has received wide attention, but it has a disadvantage in that long time may be required to complete the life test. The generalized progressive Type-II hybrid censoring scheme has recently been proposed to solve this problem. Under the censoring scheme, the time on test does not exceed a predetermined time. In this paper, we propose a robust Bayesian approach based on a hierarchical structure when the generalized progressive Type-II hybrid censored sample has a two-parameter exponential distribution. For unknown parameters, marginal posterior distributions are provided in closed forms, and their statistical properties are discussed. To examine the robustness of the proposed method, Monte Carlo simulations are conducted and a real data set is analyzed. Further, the quality and adequacy of the proposed model are evaluated in an analysis based on the real data.     

Interval estimation for exponential progressive Type-II censored step-stress accelerated life-testing

This paper derives the exact confidence intervals for the exponential step-stress accelerated life-testing model as well as the approximate confidence intervals for the k-step exponential step-stress accelerated life-testing model under progressive Type-II censoring. A Monte Carlo simulation study is carried out to examine the performance of these confidence intervals. Finally, an example is given to illustrate the proposed procedures.     

Wavelet estimation of density for censored data with censoring indicator missing at random

In this paper, we define the nonlinear wavelet estimator of density for the right censoring model with the censoring indicator missing at random (MAR), and develop its asymptotic expression for mean integrated squared error (MISE). Unlike for kernel estimator, the MISE expression of the estimator is not affected by the presence of discontinuities in the curve. Meanwhile, asymptotic normality of the estimator is established. The proposed estimator can reduce to the estimator defined by Li [Non-linear wavelet-based density estimators under random censorship. J Statist Plann Inference. 2003;117(1):35–58] when the censoring indicator MAR does not occur and a bandwidth in non-parametric estimation is close to zero. Also, we define another two nonlinear wavelet estimators of the density. A simulation is done to show the performance of the three proposed estimators.     

Estimation for the three-parameter inverse Gaussian distribution under progressive Type-II censoring

Inverse Gaussian distribution has been used widely as a model in analysing lifetime data. In this regard, estimation of parameters of two-parameter (IG2) and three-parameter inverse Gaussian (IG3) distributions based on complete and censored samples has been discussed in the literature. In this paper, we develop estimation methods based on progressively Type-II censored samples from IG3 distribution. In particular, we use the EM-algorithm, as well as some other numerical methods for determining the maximum-likelihood estimates (MLEs) of the parameters. The asymptotic variances and covariances of the MLEs from the EM-algorithm are derived by using the missing information principle. We also consider some simplified alternative estimators. The inferential methods developed are then illustrated with some numerical examples. We also discuss the interval estimation of the parameters based on the large-sample theory and examine the true coverage probabilities of these confidence intervals in case of small samples by means of Monte Carlo simulations.     

Approximate bayes estimation for mixtures of two weibull distributions under type-2 censoring

The main object of this paper is the approximate Bayes estimation of the five dimensional vector of parameters and the reliability function of a mixture of two Weibull distributions under Type-2 censoring. Under Type-2 censoring, the posterior distribution is complicated, and the integrals involved cannot be obtained in a simple closed form. In this work, Lindley's (1980) approximate form of Bayes estimation is used in the case of a mixture of two Weibull distributions under Type-2 censoring. Through Monte Carlo simulation, the root mean squared errors (RMSE's) of the Bayes estimates are computed and compared with the corresponding estimated RMSE's of the maximum likelihood estimates.     

Shrinkage testimators for the shape parameter of weibull distributin under type ii censoring

In this paper we propose some shrinkage testimators for the shape parameter of the Weibull distribution when censored samples are available and study their properties. Comparison of the testimators with Singh and Bhatkulikar (1978) and with the usual estimator, interms of mean squared error are made. It is shown that the proposed testimators     

Expected experiment times for the Weibull distribution under progressive censoring with random removals

SUMMARY This paper considers the expected experiment times for Weibull-distributed lifetimes under type II progressive censoring, with the numbers of removals being random. The formula to compute the expected experiment times is given. A detailed numerical study of this expected time is carried out for different combinations of model parameters. Furthermore, the ratio of the expected experiment time under this type of progressive censoring to the expected experiment time under complete sampling is studied.     

Optimal progressive interval censoring plan under accelerated life test with limited budget

In this paper, we investigate some inference and design problems related to multiple constant-stress accelerated life test with progressive type-I interval censoring. A Weibull lifetime distribution at each stress-level combination is considered. The scale parameter of Weibull distribution is assumed to be a log-linear function of stresses. We obtain the estimates of the unknown parameters through the method of maximum likelihood, and also derive the Fisher's information matrix. The optimal number of test units, number of inspections, and length of the inspection interval are determined under D-optimality, T-optimality, and E-optimality criteria with cost constraint. An algorithm based on nonlinear mixed-integer programming is proposed to the optimal solution. The sensitivity of the optimal solution to changes in the values of the different parameters is studied.