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.     

Exact inference for the two-parameter exponential distribution under Type-II hybrid censoring

Epstein (1954) introduced the Type-I hybrid censoring scheme as a mixture of Type-I and Type-II censoring schemes. Childs et al. (2003) introduced the Type-II hybrid censoring scheme as an alternative to Type-I hybrid censoring scheme, and provided the exact distribution of the maximum likelihood estimator of the mean of a one-parameter exponential distribution based on Type-II hybrid censored samples. The associated confidence interval also has been provided. The main aim of this paper is to consider a two-parameter exponential distribution, and to derive the exact distribution of the maximum likelihood estimators of the unknown parameters based on Type-II hybrid censored samples. The marginal distributions and the exact confidence intervals are also provided. The results can be used to derive the exact distribution of the maximum likelihood estimator of the percentile point, and to construct the associated confidence interval. Different methods are compared using extensive simulations and one data analysis has been performed for illustrative purposes.     

Generalized inverted exponential distribution under hybrid censoring

The hybrid censoring scheme is a mixture of Type-I and Type-II censoring schemes. Based on hybrid censored samples, we first derive the maximum likelihood estimators of the unknown parameters and the expected Fisher’s information matrix of the generalized inverted exponential distribution (GIED). Monte Carlo simulations are performed to study the performance of the maximum likelihood estimators. Next we consider Bayes estimation under the squared error loss function. These Bayes estimates are evaluated by applying Lindley’s approximation method, the importance sampling procedure and Metropolis–Hastings algorithm. The importance sampling technique is used to compute the highest posterior density credible intervals. Two data sets are analyzed for illustrative purposes. Finally, we discuss a method of obtaining the optimum hybrid censoring scheme.     

Parameter Estimation of Type-II Hybrid Censored Weighted Exponential Distribution

A hybrid censoring scheme is a mixture of Type-I and Type-II censoring schemes. We study the estimation of parameters of weighted exponential distribution based on Type-II hybrid censored data. By applying the EM algorithm, maximum likelihood estimators are evaluated. Using Fisher information matrix, asymptotic confidence intervals are provided. By applying Markov chain Monte Carlo techniques, Bayes estimators, and corresponding highest posterior density confidence intervals of parameters are obtained. Monte Carlo simulations are performed to compare the performances of the different methods, and one dataset is analyzed for illustrative purposes.     

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.     

A new two sample type-II progressive censoring scheme

In this paper we introduce a new type-II progressive censoring scheme for two samples. It is observed that the proposed censoring scheme is analytically more tractable than the existing joint progressive type-II censoring scheme proposed by Rasouli and Balakrishnan. The maximum likelihood estimators of the unknown parameters are obtained and their exact distributions are derived. Based on the exact distributions of the maximum likelihood estimators exact confidence intervals are also constructed. For comparison purposes we have used bootstrap confidence intervals also. One data analysis has been performed for illustrative purposes. Finally we propose some open problems.     

Inference and optimal censoring schemes for progressively censored Birnbaum–Saunders distribution

The aim of this paper is twofold. First we discuss the maximum likelihood estimators of the unknown parameters of a two-parameter Birnbaum–Saunders distribution when the data are progressively Type-II censored. The maximum likelihood estimators are obtained using the EM algorithm by exploiting the property that the Birnbaum–Saunders distribution can be expressed as an equal mixture of an inverse Gaussian distribution and its reciprocal. From the proposed EM algorithm, the observed information matrix can be obtained quite easily, which can be used to construct the asymptotic confidence intervals. We perform the analysis of two real and one simulated data sets for illustrative purposes, and the performances are quite satisfactory. We further propose the use of different criteria to compare two different sampling schemes, and then find the optimal sampling scheme for a given criterion. It is observed that finding the optimal censoring scheme is a discrete optimization problem, and it is quite a computer intensive process. We examine one sub-optimal censoring scheme by restricting the choice of censoring schemes to one-step censoring schemes as suggested by Balakrishnan (2007), which can be obtained quite easily. We compare the performances of the sub-optimal censoring schemes with the optimal ones, and observe that the loss of information is quite insignificant.     

Statistical Analysis of the Log-Normal Distribution under Type-II Progressive Hybrid Censoring Schemes

In this article, we obtain the maximum likelihood estimators (MLEs) and approximate maximum likelihood estimators (AMLEs) of the parameters, from a two-parameter log-normal distribution based on the adaptive Type-II progressive hybrid censoring scheme, which was introduced by Ng et al. (2009 Ng , H. K. T. , Kundu , D. , Chan , P. S. ( 2009 ). Statistical analysis of exponential lifetimes under an adaptive Type-II progressively censoring scheme . Naval Research Logistics 56 : 687 – 698 .[Crossref], [Web of Science ®] , [Google Scholar]) for life testing or reliability experiment. In order to compare the results, we calculate corresponding estimators of the Type-II progressive hybrid censoring scheme. In particular, we provide computational formulas of the expected total test time and the expected number of failures for each scheme. We also compute the observed Fisher information matrix and use them to obtain the asymptotic confidence intervals. A simulation study carries out to evaluate the bias and mean square error of the MLEs and AMLEs from the two above-mentioned schemes. Finally, we present a numerical example to illustrate the methods of inference discussed here.     

Point and Interval Estimation for a Generalized Logistic Distribution Under Progressive Type II Censoring

The likelihood equations based on a progressively Type II censored sample from a Type I generalized logistic distribution do not provide explicit solutions for the location and scale parameters. We present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We examine numerically the bias and variance of these estimators and show that these estimators are as efficient as the maximum likelihood estimators (MLEs). The probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are shown to be unsatisfactory, especially when the effective sample size is small. Therefore we suggest using unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. A wide range of sample sizes and progressive censoring schemes have been considered in this study. Finally, we present a numerical example to illustrate the methods of inference developed here.     

Aalen's linear model for doubly censored data

Double censoring often occurs in registry studies when left censoring is present in addition to right censoring. In this work, we examine estimation of Aalen's nonparametric regression coefficients based on doubly censored data. We propose two estimation techniques. The first type of estimators, including ordinary least squared (OLS) estimator and weighted least squared (WLS) estimators, are obtained using martingale arguments. The second type of estimator, the maximum likelihood estimator (MLE), is obtained via expectation-maximization (EM) algorithms that treat the survival times of left censored observations as missing. Asymptotic properties, including the uniform consistency and weak convergence, are established for the MLE. Simulation results demonstrate that the MLE is more efficient than the OLS and WLS estimators.     

Exact likelihood inference for two exponential populations based on a joint generalized Type-I hybrid censored sample

Following the work of Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring. Comm Statist Theory Methods. 1988;17:1857–1870], several results have been developed regarding the exact likelihood inference of exponential parameters based on different forms of censored samples. In this paper, the conditional maximum likelihood estimators (MLEs) of two exponential mean parameters are derived under joint generalized Type-I hybrid censoring on the two samples. The moment generating functions (MGFs) and the exact densities of the conditional MLEs are obtained, using which exact confidence intervals are then developed for the model parameters. We also derive the means, variances, and mean squared errors of these estimates. An efficient computational method is developed based on the joint MGF. Finally, an example is presented to illustrate the methods of inference developed 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.     

Statistical analysis of competing risks model from Marshall–Olkin extended Chen distribution under adaptive progressively interval censoring with random removals

In this paper, a new censoring scheme named by adaptive progressively interval censoring scheme is introduced. The competing risks data come from Marshall–Olkin extended Chen distribution under the new censoring scheme with random removals. We obtain the maximum likelihood estimators of the unknown parameters and the reliability function by using the EM algorithm based on the failure data. In addition, the bootstrap percentile confidence intervals and bootstrap-t confidence intervals of the unknown parameters are obtained. To test the equality of the competing risks model, the likelihood ratio tests are performed. Then, Monte Carlo simulation is conducted to evaluate the performance of the estimators under different sample sizes and removal schemes. Finally, a real data set is analyzed for illustration purpose.     

Approximate MLE for the scale parameter of the generalized exponential distribution under random censoring

In this paper, we consider the maximum likelihood estimator (MLE) of the scale parameter of the generalized exponential (GE) distribution based on a random censoring model. We assume the censoring distribution also follows a GE distribution. Since the estimator does not provide an explicit solution, we propose a simple method of deriving an explicit estimator by approximating the likelihood function. In order to compare the performance of the estimators, Monte Carlo simulation is conducted. The results show that the MLE and the approximate MLE are almost identical in terms of bias and variance.     

On the estimation of the extreme value and normal distribution parameters based on progressive type-II hybrid-censored data

A progressive hybrid censoring scheme is a mixture of type-I and type-II progressive censoring schemes. In this paper, we mainly consider the analysis of progressive type-II hybrid-censored data when the lifetime distribution of the individual item is the normal and extreme value distributions. Since the maximum likelihood estimators (MLEs) of these parameters cannot be obtained in the closed form, we propose to use the expectation and maximization (EM) algorithm to compute the MLEs. Also, the Newton–Raphson method is used to estimate the model parameters. The asymptotic variance–covariance matrix of the MLEs under EM framework is obtained by Fisher information matrix using the missing information and asymptotic confidence intervals for the parameters are then constructed. This study will end up with comparing the two methods of estimation and the asymptotic confidence intervals of coverage probabilities corresponding to the missing information principle and the observed information matrix through a simulation study, illustrated examples and real data analysis.     

Reliability analysis for stress-strength model from a general family of truncated distributions under censored data

Abstract

Under progressive Type-II censoring, inference of stress-strength reliability (SSR) is studied for a general family of lower truncated distributions. When the lifetime models of the strength and stress variables have arbitrary and common parameters, maximum likelihood and pivotal quantities based generalized estimators of SSR are established, respectively. Confidence intervals are constructed based on generalized pivotal quantities and bootstrap technique under different parameter cases as well. In addition, to compare the equivalence of the strength and stress parameters, likelihood ratio testing of interested parameters is provided as a complementary. Simulation studies and two real-life data examples are provided to investigate the performance of proposed methods.     

On progressively censored inverted exponentiated Rayleigh distribution

In this paper, we discuss a progressively censored inverted exponentiated Rayleigh distribution. Estimation of unknown parameters is considered under progressive censoring using maximum likelihood and Bayesian approaches. Bayes estimators of unknown parameters are derived with respect to different symmetric and asymmetric loss functions using gamma prior distributions. An importance sampling procedure is taken into consideration for deriving these estimates. Further highest posterior density intervals for unknown parameters are constructed and for comparison purposes bootstrap intervals are also obtained. Prediction of future observations is studied in one- and two-sample situations from classical and Bayesian viewpoint. We further establish optimum censoring schemes using Bayesian approach. Finally, we conduct a simulation study to compare the performance of proposed methods and analyse two real data sets for illustration purposes.     

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.