Generalized inverted exponential distribution under progressive first-failure censoring

This article deals with progressive first-failure censoring, which is a generalization of progressive censoring. We derive maximum likelihood estimators of the unknown parameters and reliability characteristics of generalized inverted exponential distribution using progressive first-failure censored samples. The asymptotic confidence intervals and coverage probabilities for the parameters are obtained based on the observed Fisher's information matrix. Bayes estimators of the parameters and reliability characteristics under squared error loss function are obtained using the Lindley approximation and importance sampling methods. Also, highest posterior density credible intervals for the parameters are computed using importance sampling procedure. A Monte Carlo simulation study is conducted to analyse the performance of the estimators derived in the article. A real data set is discussed for illustration purposes. Finally, an optimal censoring scheme has been suggested using different optimality criteria.     

Bayesian Analysis of Different Hybrid and Progressive Life Tests

Type-I and Type-II censoring schemes are the widely used censoring schemes available for life testing experiments. A mixture of Type-I and Type-II censoring schemes is known as a hybrid censoring scheme. Different hybrid censoring schemes have been introduced in recent years. In the last few years, a progressive censoring scheme has also received considerable attention. In this article, we mainly consider the Bayesian inference of the unknown parameters of two-parameter exponential distribution under different hybrid and progressive censoring schemes. It is observed that in general the Bayes estimate and the associated credible interval of any function of the unknown parameters, cannot be obtained in explicit form. We propose to use the Monte Carlo sampling procedure to compute the Bayes estimate and also to construct the associated credible interval. Monte Carlo Simulation experiments have been performed to see the effectiveness of the proposed method in case of Type-I hybrid censored samples. The performances are quite satisfactory. One data analysis has been performed for illustrative purposes.     

Exact confidence bounds for an exponential parameter under hybrid censoring

Consider a life testing experiment in which n units are put on test, successive failure times are recorded, and the observation is terminated either at a specified number r of failures or a specified time T whichever is reached first. This mixture of type I and type II censoring schemes, called hybrid censoring, is of wide use. Under this censoring scheme and the assumption of an exponential life distribution, the distribution of the maximum likelihood estimator of the mean life 6 is derived. It is then used to construct an exact lower confidence bound for θ.     

Exact confidence bounds for an exponential parameter under hybrid censoring

Consider a life testing experiment in which n units are put on test, successive failure times are recorded, and the observation is terminated either at a specified number r of failures or a specified time T whichever is reached first. This mixture of type I and type II censoring schemes, called hybrid censoring, is of wide use. Under this censoring scheme and the assumption of an exponential life distribution, the distribution of the maximum likelihood estimator of the mean life θ is derived. It is then used to construct an exact lower confidence bound for θ.     

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.     

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.     

Estimation based on progressive first-failure censoring from exponentiated exponential distribution

In this paper, point and interval estimations for the parameters of the exponentiated exponential (EE) distribution are studied based on progressive first-failure-censored data. The Bayes estimates are computed based on squared error and Linex loss functions and using Markov Chain Monte Carlo (MCMC) algorithm. Also, based on this censoring scheme, approximate confidence intervals for the parameters of EE distribution are developed. Monte Carlo simulation study is carried out to compare the performances of the different methods by computing the estimated risks (ERs), as well as Akaike's information criteria (AIC) and Bayesian information criteria (BIC) of the estimates. Finally, a real data set is introduced and analyzed using EE and Weibull distributions. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic to emphasize that the EE model fits the data with the same efficiency as the other model. Point and interval estimation of all parameters are studied based on this real data set as illustrative example.     

Exact inference on multiple exponential populations under a joint type-II progressive censoring scheme

ABSTRACT

Recently Mondal and Kundu [Mondal S, Kundu D. A new two sample type-II progressive censoring scheme. Commun Stat Theory Methods. 2018. doi:10.1080/03610926.2018.1472781] introduced a Type-II progressive censoring scheme for two populations. In this article, we extend the above scheme for more than two populations. The aim of this paper is to study the statistical inference under the multi-sample Type-II progressive censoring scheme, when the underlying distributions are exponential. We derive the maximum likelihood estimators (MLEs) of the unknown parameters when they exist and find out their exact distributions. The stochastic monotonicity of the MLEs has been established and this property can be used to construct exact confidence intervals of the parameters via pivoting the cumulative distribution functions of the MLEs. The distributional properties of the ordered failure times are also obtained. The Bayesian analysis of the unknown model parameters has been provided. The performances of the different methods have been examined by extensive Monte Carlo simulations. We analyse two data sets for illustrative purposes.     

Fisher information in generalized progressive hybrid-censored data

This article provides a simple expression of the Fisher information matrix about the unknown parameter(s) of the underlying lifetime model under the generalized progressive hybrid censoring scheme. The expressions of the expected number of failures and the expected duration of life test are also derived. Exponential and Weibull lifetime models are considered for numerical illustrations. Finally, Fisher information-based optimal schemes are discussed for the Weibull lifetime model.     

Exact inference for competing risks model with generalized type-I hybrid censored exponential data

Recently, exact inference under hybrid censoring scheme has attracted extensive attention in the field of reliability analysis. However, most of the authors neglect the possibility of competing risks model. This paper mainly discusses the exact likelihood inference for the analysis of generalized type-I hybrid censoring data with exponential competing failure model. Based on the maximum likelihood estimates for unknown parameters, we establish the exact conditional distribution of parameters by conditional moment generating function, and then obtain moment properties as well as exact confidence intervals (CIs) for parameters. Furthermore, approximate CIs are constructed by asymptotic distribution and bootstrap method as well. We also compare their performances with exact method through the use of Monte Carlo simulations. And finally, a real data set is analysed to illustrate the validity of all the methods developed here.     

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.     

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.     

Progressive censoring with fixed censoring times

In this paper, we revisit the progressive Type-I censoring scheme as it has originally been introduced by Cohen [Progressively censored samples in life testing. Technometrics. 1963;5(3):327–339]. In fact, original progressive Type-I censoring proceeds as progressive Type-II censoring but with fixed censoring times instead of failure time based censoring times. Apparently, a time truncation has been added to this censoring scheme by interpreting the final censoring time as a termination time. Therefore, not much work has been done on Cohens's original progressive censoring scheme with fixed censoring times. Thus, we discuss distributional results for this scheme and establish exact distributional results in likelihood inference for exponentially distributed lifetimes. In particular, we obtain the exact distribution of the maximum likelihood estimator (MLE). Further, the stochastic monotonicity of the MLE is verified in order to construct exact confidence intervals for both the scale parameter and the reliability.     

Hybrid censoring schemes with exponential failure distribution

The mixture of Type I and Type I1 censoring schemes, called the hybrid censoring, is quite important in life–testing experiments. Epstein(1954, 1960) introduced this testing scheme and proposed a two–sided confidence interval to estimate the mean lifetime, θ, when the underlying lifetime distribution is assumed to be exponential. There are some two–sided confidence intervals and credible intervals proposed by Fairbanks et al. (1982) and Draper and Guttman (1987) respectively. In this paper we obtain the exact two–sided confidence interval of θ following the approach of Chen and Bhattacharya (1988). We also obtain the asymptotic confidence intervals in the Hybrid censoring case. It is important to observe that the results for Type I and Type II censoring schemes can be obtained as particular cases of the Hybrid censoring scheme. We analyze one data set and compare different methods by Monte Carlo simulations.     

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.     

Development of life-test sampling plans for exponential distributions based on accelerated life testing

This paper presents a procedure for developing life-test sampling plans for exponential distributions based upon accelerated life testing(ALT). Type II censoring is assumed at each overstress level. The derived test statistic is shown to be a quotient of two independent random variables, each of which is a rational power of a Chi-square random variable. The distribution of the test statistic is characterized by the H-function, which can be numerically evaluated to obtain desired sampling plans.     

Exact distribution of the MLEs of the parameters and of the quantiles of two-parameter exponential distribution under hybrid censoring

Epstein [Truncated life tests in the exponential case, Ann. Math. Statist. 25 (1954), pp. 555–564] introduced a hybrid censoring scheme (called Type-I hybrid censoring) and Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods 17 (1988), pp. 1857–1870] derived the exact distribution of the maximum-likelihood estimator (MLE) of the mean of a scaled exponential distribution based on a Type-I hybrid censored sample. Childs et al. [Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math. 55 (2003), pp. 319–330] provided an alternate simpler expression for this distribution, and also developed analogous results for another hybrid censoring scheme (called Type-II hybrid censoring). The purpose of this paper is to derive the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored samples. The marginal distributions are derived and exact confidence bounds for the parameters are obtained. The results are also used to derive the exact distribution of the MLE of the pth quantile, as well as the corresponding confidence bounds. These exact confidence intervals are then compared with parametric bootstrap confidence intervals in terms of coverage probabilities. Finally, we present some numerical examples to illustrate the methods of inference developed here.     

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.     

Efficient Bayesian sampling plans for exponential distributions with random censoring

This paper considers Bayesian sampling plans for exponential distribution with random censoring. The efficient Bayesian sampling plan for a general loss function is derived. This sampling plan possesses the property that it may make decisions prior to the end of the life test experiment, and its decision function is the same as the Bayes decision function which makes decisions based on data collected at the end of the life test experiment. Compared with the optimal Bayesian sampling plan of Chen et al. (2004), the efficient Bayesian sampling plan has the smaller Bayes risk due to the less duration time of life test experiment. Computations of the efficient Bayes risks for the conjugate prior are given. Numerical comparisons between the proposed efficient Bayesian sampling plan and the optimal Bayesian sampling plan of Chen et al. (2004) under two special decision losses, including the quadratic decision loss, are provided. Numerical results also demonstrate that the performance of the proposed efficient sampling plan is superior to that of the optimal sampling plan by Chen et al. (2004).