Inference for Weibull distribution based on progressively Type-II hybrid censored data

Progressive Type-II hybrid censoring is a mixture of progressive Type-II and hybrid censoring schemes. In this paper, we discuss the statistical inference on Weibull parameters when the observed data are progressively Type-II hybrid censored. We derive the maximum likelihood estimators (MLEs) and the approximate maximum likelihood estimators (AMLEs) of the Weibull parameters. We then use the asymptotic distributions of the maximum likelihood estimators to construct approximate confidence intervals. Bayes estimates and the corresponding highest posterior density credible intervals of the unknown parameters are obtained under suitable priors on the unknown parameters and also by using the Gibbs sampling procedure. Monte Carlo simulations are then performed for comparing the confidence intervals based on all those different methods. Finally, one data set is analyzed for illustrative purposes.     

Inference on constant stress accelerated life tests for log-location-scale lifetime distributions with type-I hybrid censoring

In this paper, we consider a constant stress accelerated life test terminated by a hybrid Type-I censoring at the first stress level. The model is based on a general log-location-scale lifetime distribution with mean life being a linear function of stress and with constant scale. We obtain the maximum likelihood estimators (MLE) and the approximate maximum likelihood estimators (AMLE) of the model parameters. Approximate confidence intervals, likelihood ratio tests and two bootstrap methods are used to construct confidence intervals for the unknown parameters of the Weibull and lognormal distributions using the MLEs. Finally, a simulation study and two illustrative examples are provided to demonstrate the performance of the developed inferential methods.     

Likelihood inference for the component lifetime distribution based on progressively censored parallel systems data

Point and interval estimators for the scale parameter of the component lifetime distribution of a k-component parallel system are obtained when the component lifetimes are assumed to be independently and identically exponentially distributed. We prove that the maximum likelihood estimator of the scale parameter based on progressively Type-II censored system lifetimes is unique and can be obtained by a fixed-point iteration procedure. In particular, we illustrate that the Newton–Raphson method does not converge for any initial value. Furthermore, exact confidence intervals are constructed by a transformation using normalized spacings and other component lifetime distributions including Weibull distribution are discussed.     

Estimation of the extended Weibull parameters and acceleration factors in the step-stress accelerated life tests under an adaptive progressively hybrid censoring data

ABSTRACT

Based on the tampered failure rate model under the adaptive Type-II progressively hybrid censoring data, we discuss the maximum likelihood estimators of the unknown parameters and acceleration factors in the general step-stress accelerated life tests in this paper. We also construct the exact and unique confidence interval for the extended Weibull shape parameter. In the numerical analysis, we describe the simulation procedures to obtain the adaptive Type-II progressively hybrid censoring data in the step-stress accelerated life tests and present an experimental data to illustrate the performance of the estimators.     

Statistical inference for two exponential populations under joint progressive type-I censored scheme

In this article, we introduce a new scheme called joint progressive type-I (JPC-I) censored and as a special case, joint type-I censored scheme. Bayesian and non Bayesian estimators have been obtained for two exponential populations under both JPC-I censored scheme and joint type-I censored. The maximum likelihood estimators of the parameters, the asymptotic variance covariance matrix, have been obtained. Bayes estimators have been developed under squared error loss function using independent gamma prior distributions. Moreover, approximate confidence region based on the asymptotic normality of the maximum likelihood estimators and credible confidence region from a Bayesian viewpoint are also discussed and compared with two Bootstrap confidence regions. A numerical illustration for these new results is given.     

On hybrid censored Weibull distribution

A hybrid censoring is a mixture of Type-I and Type-II censoring schemes. This article presents the statistical inferences on Weibull parameters when the data are hybrid censored. The maximum likelihood estimators (MLEs) and the approximate maximum likelihood estimators are developed for estimating the unknown parameters. Asymptotic distributions of the MLEs are used to construct approximate confidence intervals. Bayes estimates and the corresponding highest posterior density credible intervals of the unknown parameters are obtained under suitable priors on the unknown parameters and using the Gibbs sampling procedure. The method of obtaining the optimum censoring scheme based on the maximum information measure is also developed. Monte Carlo simulations are performed to compare the performances of the different methods and one data set is analyzed for illustrative purposes.     

A class of absolutely continuous bivariate distributions

Block and Basu bivariate exponential distribution is one of the most popular absolutely continuous bivariate distributions. Extensive work has been done on the Block and Basu bivariate exponential model over the past several decades. Interestingly it is observed that the Block and Basu bivariate exponential model can be extended to the Weibull model also. We call this new model as the Block and Basu bivariate Weibull model. We consider different properties of the Block and Basu bivariate Weibull model. The Block and Basu bivariate Weibull model has four unknown parameters and the maximum likelihood estimators cannot be obtained in closed form. To compute the maximum likelihood estimators directly, one needs to solve a four dimensional optimization problem. We propose to use the EM algorithm for computing the maximum likelihood estimators of the unknown parameters. The proposed EM algorithm can be carried out by solving one non-linear equation at each EM step. Our method can be also used to compute the maximum likelihood estimators for the Block and Basu bivariate exponential model. One data analysis has been preformed for illustrative purpose.     

Statistical analysis of dependent competing risks model in accelerated life testing under progressively hybrid censoring using copula function

This article considers statistical analysis of dependent competing risks model from Weibull distribution in accelerated life testing, in which copula function is used to examine the dependence structure between competing failure modes. We derive the maximum likelihood estimates, the approximate, and Bootstrap confidence intervals of the parameters. The effects of different dependence structures on the estimates of parameters are investigated. The simulation is given to compare the performance of the estimates when the competing failure modes are dependent with those when the failure modes are independent. Finally, one dataset was used for illustrative purpose in conclusion.     

Exact Likelihood Inference for Two Exponential Populations Under Joint Progressive Type-II Censoring

Comparative lifetime experiments are of great importance when the interest is in ascertaining the relative merits of two competing products with regard to their reliability. In this article, we consider two exponential populations and when joint progressive Type-II censoring is implemented on the two samples. We then derive the moment generating functions and the exact distributions of the maximum likelihood estimators (MLEs) of the mean lifetimes of the two exponential populations under such a joint progressive Type-II censoring. We then discuss the exact lower confidence bounds, exact confidence intervals, and simultaneous confidence regions. Next, we discuss the corresponding approximate results based on the asymptotic normality of the MLEs as well as those based on the Bayesian method. All these confidence intervals and regions are then compared by means of Monte Carlo simulations with those obtained from bootstrap methods. Finally, an illustrative example is presented in order to illustrate all the methods of inference discussed here.     

ESTIMATION OF EXPONENTIATED WEIBULL SHAPE PARAMETERS UNDER LINEX LOSS FUNCTION

ABSTRACT

The paper deals with Bayes estimation of the exponentiated Weibull shape parameters under linex loss function when independent non-informative type of priors are available for the parameters. Generalized maximum likelihood estimators have also been obtained. Performances of the proposed Bayes estimator, generalized maximum likelihood estimators, posterior mean (i.e., Bayes estimator under squared error loss function) and maximum likelihood estimators have been studied on the basis of their risks under linex loss function. The comparison is based on a simulation study because the expressions for risk functions of these estimators cannot be obtained in nice closed forms.     

LASSO and shrinkage estimation in Weibull censored regression models

In this paper we address the problem of estimating a vector of regression parameters in the Weibull censored regression model. Our main objective is to provide natural adaptive estimators that significantly improve upon the classical procedures in the situation where some of the predictors may or may not be associated with the response. In the context of two competing Weibull censored regression models (full model and candidate submodel), we consider an adaptive shrinkage estimation strategy that shrinks the full model maximum likelihood estimate in the direction of the submodel maximum likelihood estimate. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Further, we consider a LASSO type estimation strategy and compare the relative performance with the shrinkage estimators. Monte Carlo simulations reveal that when the true model is close to the candidate submodel, the shrinkage strategy performs better than the LASSO strategy when, and only when, there are many inactive predictors in the model. Shrinkage and LASSO strategies are applied to a real data set from Veteran's administration (VA) lung cancer study to illustrate the usefulness of the procedures in practice.     

Statistical Inference of Type-II Progressively Hybrid Censored Data with Weibull Lifetimes

In this article, we discuss the maximum likelihood estimators and approximate maximum likelihood estimators of the parameters of the Weibull distribution with two different progressively hybrid censoring schemes. We also present the associated expressions of the expected total test time and the expected effective sample size which will be useful for experimental planning purpose. Finally, the efficiency of the point estimation of the parameters based on the two progressive hybrid censoring schemes are compared and the merits of each censoring scheme are discussed.     

Interval estimation of the largest reliability of k weibull populations

A procedure is given for obtaining a random width confidence interval for the largest reliability of k Weibull populations. The procedure does not identify the populations for which the reliability would be a maximum. The maximum likelihood estimators or the simplified linear estimators of the reliability based on type II censored data are used. The cases considered include unknown shape parameters being equal or unequal. Simultaneous confidence intervals for the k reliabilities are also obtained. Tables for the lower and upper limits in selected cases are constructed using Monte Carlo methods.     

An extended Maxwell distribution: Properties and applications

In this article, we propose an extension of the Maxwell distribution, so-called the extended Maxwell distribution. This extension is evolved by using the Maxwell-X family of distributions and Weibull distribution. We study its fundamental properties such as hazard rate, moments, generating functions, skewness, kurtosis, stochastic ordering, conditional moments and moment generating function, hazard rate, mean and variance of the (reversed) residual life, reliability curves, entropy, etc. In estimation viewpoint, the maximum likelihood estimation of the unknown parameters of the distribution and asymptotic confidence intervals are discussed. We also obtain expected Fisher’s information matrix as well as discuss the existence and uniqueness of the maximum likelihood estimators. The EMa distribution and other competing distributions are fitted to two real datasets and it is shown that the distribution is a good competitor to the compared distributions.     

Bayes estimation for the Block and Basu bivariate and multivariate Weibull distributions

Block and Basu bivariate exponential distribution is one of the most popular absolute continuous bivariate distributions. Recently, Kundu and Gupta [A class of absolute continuous bivariate distributions. Statist Methodol. 2010;7:464–477] introduced Block and Basu bivariate Weibull (BBBW) distribution, which is a generalization of the Block and Basu bivariate exponential distribution, and provided the maximum likelihood estimators using EM algorithm. In this paper, we consider the Bayesian inference of the unknown parameters of the BBBW distribution. The Bayes estimators are obtained with respect to the squared error loss function, and the prior distributions allow for prior dependence among the unknown parameters. Prior independence also can be obtained as a special case. It is observed that the Bayes estimators of the unknown parameters cannot be obtained in explicit forms. We propose to use the importance sampling technique to compute the Bayes estimates and also to construct the associated highest posterior density credible intervals. The analysis of two data sets has been performed for illustrative purposes. The performances of the proposed estimators are quite satisfactory. Finally, we generalize the results for the multivariate case.     

Estimation of parameters of inverse Weibull distribution and application to multi-component stress-strength model

The problem of estimation of the parameters of two-parameter inverse Weibull distributions has been considered. We establish existence and uniqueness of the maximum likelihood estimators of the scale and shape parameters. We derive Bayes estimators of the parameters under the entropy loss function. Hierarchical Bayes estimator, equivariant estimator and a class of minimax estimators are derived when shape parameter is known. Ordered Bayes estimators using information about second population are also derived. We investigate the reliability of multi-component stress-strength model using classical and Bayesian approaches. Risk comparison of the classical and Bayes estimators is done using Monte Carlo simulations. Applications of the proposed estimators are shown using real data sets.     

Bayesian estimation for the exponentiated Weibull model under Type II progressive censoring

Based on progressive Type II censored samples, we have derived the maximum likelihood and Bayes estimators for the two shape parameters and the reliability function of the exponentiated Weibull lifetime model. We obtained Bayes estimators using both the symmetric and asymmetric loss functions via squared error loss and linex loss functions. This was done with respect to the conjugate priors for two shape parameters. We used an approximation based on the Lindley (Trabajos de Stadistca 21, 223–237, 1980) method for obtaining Bayes estimates under these loss functions. We made comparisons between these estimators and the maximum likelihood estimators using a Monte Carlo simulation study.     

Empirical likelihood inference for partial functional linear model with missing responses

In this paper, we consider the empirical likelihood inferences of the partial functional linear model with missing responses. Two empirical log-likelihood ratios of the parameters of interest are constructed, and the corresponding maximum empirical likelihood estimators of parameters are derived. Under some regularity conditions, we show that the proposed two empirical log-likelihood ratios are asymptotic standard Chi-squared. Thus, the asymptotic results can be used to construct the confidence intervals/regions for the parameters of interest. We also establish the asymptotic distribution theory of corresponding maximum empirical likelihood estimators. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals. An example of real data is also used to illustrate our proposed methods.     

Bivariate Weibull regression model based on censored samples

The most natural parametric distribution to consider is the Weibull model because it allows for both the proportional hazard model and accelerated failure time model. In this paper, we propose a new bivariate Weibull regression model based on censored samples with common covariates. There are some interesting biometrical applications which motivate to study bivariate Weibull regression model in this particular situation. We obtain maximum likelihood estimators for the parameters in the model and test the significance of the regression parameters in the model. We present a simulation study based on 1000 samples and also obtain the power of the test statistics.     

Using BBPSO Algorithm to Estimate the Weibull Parameters with Censored Data

This article proposes the maximum likelihood estimates based on bare bones particle swarm optimization (BBPSO) algorithm for estimating the parameters of Weibull distribution with censored data, which is widely used in lifetime data analysis. This approach can produce more accuracy of the parameter estimation for the Weibull distribution. Additionally, the confidence intervals for the estimators are obtained. The simulation results show that the BB PSO algorithm outperforms the Newton–Raphson method in most cases in terms of bias, root mean square of errors, and coverage rate. Two examples are used to demonstrate the performance of the proposed approach. The results show that the maximum likelihood estimates via BBPSO algorithm perform well for estimating the Weibull parameters with censored data.