Conservative Likelihood Inference for Type I Censored Samples from the Log-Location-Scale Distributions

A new method, theoretically justified, is proposed to overcome difficulties in analysing reliability data coming from Type I censored samples. The article shows that, by means of Monte Carlo simulations, it is possible to obtain quasi-exact likelihood estimator properties and conservative confidence intervals for log-location-scale distributions. In the case of the exponential distribution, comparisons with the exact estimator properties show that the Monte Carlo approach allows to calculate the properties with very good accuracy. Finally, for the exponential distribution it is demonstrated that, if the number of failures can only be different from zero, confidence intervals based on the asymptotic properties of the likelihood estimators may give statistically meaningless results in the case of small sample size (3–10) and low probability of failure (.05–.20).     

Analysis of hybrid censored competing risks data

In this paper, we consider the analysis of hybrid censored competing risks data, based on Cox's latent failure time model assumptions. It is assumed that lifetime distributions of latent causes of failure follow Weibull distribution with the same shape parameter, but different scale parameters. Maximum likelihood estimators (MLEs) of the unknown parameters can be obtained by solving a one-dimensional optimization problem, and we propose a fixed-point type algorithm to solve this optimization problem. Approximate MLEs have been proposed based on Taylor series expansion, and they have explicit expressions. Bayesian inference of the unknown parameters are obtained based on the assumption that the shape parameter has a log-concave prior density function, and for the given shape parameter, the scale parameters have Beta–Gamma priors. We propose to use Markov Chain Monte Carlo samples to compute Bayes estimates and also to construct highest posterior density credible intervals. Monte Carlo simulations are performed to investigate the performances of the different estimators, and two data sets have been analysed for illustrative purposes.     

Exact Inference for a Simple Step-stress Model with Generalized Type-I Hybrid Censored Data from the Exponential Distribution

In this article, the simple step-stress model is considered based on generalized Type-I hybrid censored data from the exponential distribution. The maximum likelihood estimators (MLEs) of the unknown parameters are derived assuming a cumulative exposure model. We then derive the exact distributions of the MLEs of the parameters using conditional moment generating functions. The Bayesian estimators of the parameters are derived and then compared with the MLEs. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs, Bayesian, and the parametric bootstrap methods. The problem of determining the optimal stress-changing point is discussed and the MLEs of the pth quantile and reliability functions at the use condition are obtained. Finally, Monte Carlo simulation and some numerical results are presented for illustrating all the inferential methods developed here.     

Statistical analysis of middle censored competing risks data with exponential distribution

In this paper, we consider some problems of estimation and reconstruction based on middle censored competing risks data. It is assumed that the lifetime distributions of the latent failure times are independent and exponential distributed with different parameters and also that the censoring mechanism is independent. The maximum likelihood estimators (MLEs) of the unknown parameters are obtained. We then use the asymptotic distribution of the MLEs to construct approximate confidence intervals. Based on gamma priors, Lindley's approximation method is applied to obtain the Bayesian estimates of the unknown parameters under squared error loss function. Since it is not possible to construct the credible intervals, we propose and implement the Gibbs sampling technique to construct the credible intervals. Several point reconstructors for failure time of censored units are provided. Finally, a simulation study is given by Monte-Carlo simulations to evaluate the performances of the different methods and a data set is analysed to illustrate the proposed procedures.     

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.     

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.     

On adaptive progressively Type-II censored competing risks data

In this article, a competing risks model based on exponential distributions is considered under the adaptive Type-II progressively censoring scheme introduced by Ng et al. [2009, Naval Research Logistics 56:687-698], for life testing or reliability experiment. Moreover, we assumed that some causes of failures are unknown. The maximum likelihood estimators (MLEs) of unknown parameters are established. The exact conditional and the asymptotic distributions of the obtained estimators are derived to construct the confidence intervals as well as the two different bootstraps of different unknown parameters. Under suitable priors on the unknown parameters, Bayes estimates and the corresponding two sides of Bayesian probability intervals are obtained. Also, for the purpose of evaluating the average bias and mean square error of the MLEs, and comparing the confidence intervals based on all mentioned methods, a simulation study was carried out. Finally, we present one real dataset to conduct the proposed methods.     

On the simple step-stress model for two-parameter exponential distribution

In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes.     

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.     

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.     

Analysis of simple step-stress model in presence of competing risks

ABSTRACT

In this article, we consider a simple step-stress life test in the presence of exponentially distributed competing risks. It is assumed that the stress is changed when a pre-specified number of failures takes place. The data is assumed to be Type-II censored. We obtain the maximum likelihood estimators of the model parameters and the exact conditional distributions of the maximum likelihood estimators. Based on the conditional distribution, approximate confidence intervals (CIs) of unknown parameters have been constructed. Percentile bootstrap CIs of model parameters are also provided. Optimal test plan is addressed. We perform an extensive simulation study to observe the behaviour of the proposed method. The performances are quite satisfactory. Finally we analyse two data sets for illustrative purposes.     

Analysis of incomplete data in the presence of dependent competing risks from Marshall–Olkin bivariate Weibull distribution under progressive hybrid censoring

This article considers the statistical analysis of dependent competing risks model with incomplete data under Type-I progressive hybrid censored condition using a Marshall–Olkin bivariate Weibull distribution. Based on the expectation maximum algorithm, maximum likelihood estimators for the unknown parameters are obtained, and the missing information principle is used to obtain the observed information matrix. As the maximum likelihood approach may fail when the available information is insufficient, Bayesian approach incorporated with auxiliary variables is developed for estimating the parameters of the model, and Monte Carlo method is employed to construct the highest posterior density credible intervals. The proposed method is illustrated through a numerical example under different progressive censoring schemes and masking probabilities. Finally, a real data set is analyzed for illustrative purposes.     

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.     

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.     

Progressively Type-II censored competing risks data for exponential distributions based on sequential order statistics

We consider the progressively Type-II censored competing risks model based on sequential order statistics. It is assumed that the latent failure times are independent and the failure of each unit influences the lifetime distributions of the latent failure times of surviving units. We provide explicit expressions for the likelihood function of the available data under the conditional proportional hazard rate (CPHR) and the power trend conditional proportional hazard rate (PTCPHR) models. Under CPHR and PTCPHR models and assumption that the baseline distributions of the latent failure times are exponential, classical and Bayesian estimates of the unknown parameters are provided. Monte Carlo simulations are then performed for illustrative purposes. Finally, two datasets are analyzed.     

Estimation in Maxwell distribution with randomly censored data

In many practical situations, complete data are not available in lifetime studies. Many of the available observations are right censored giving survival information up to a noted time and not the exact failure times. This constitutes randomly censored data. In this paper, we consider Maxwell distribution as a survival time model. The censoring time is also assumed to follow a Maxwell distribution with a different parameter. Maximum likelihood estimators and confidence intervals for the parameters are derived with randomly censored data. Bayes estimators are also developed with inverted gamma priors and generalized entropy loss function. A Monte Carlo simulation study is performed to compare the developed estimation procedures. A real data example is given at the end of the study.     

Estimation of generalized exponential distribution based on an adaptive progressively type-II censored sample

In this paper, based on an adaptive Type-II progressively censored sample from the generalized exponential distribution, the maximum likelihood and Bayesian estimators are derived for the unknown parameters as well as the reliability and hazard functions. Also, the approximate confidence intervals of the unknown parameters, and the reliability and hazard functions are calculated. Markov chain Monte Carlo method is applied to carry out a Bayesian estimation procedure and in turn calculate the credible intervals. Moreover, results from simulation studies assessing the performance of our proposed method are included. Finally, an illustrative example using real data set is presented for illustrating all the inferential procedures developed here.     

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.     

Progressively Type-I interval censored competing risks data for the proportional hazards family

In this article, we consider some problems of estimation and prediction when progressive Type-I interval censored competing risks data are from the proportional hazards family. The maximum likelihood estimators of the unknown parameters are obtained. Based on gamma priors, the Lindely's approximation and importance sampling methods are applied to obtain Bayesian estimators under squared error and linear–exponential loss functions. Several classical and Bayesian point predictors of censored units are provided. Also, based on given producer's and consumer's risks accepting sampling plans are considered. Finally, the simulation study is given by Monte Carlo simulations to evaluate the performances of the different methods.     

Exact inference for a simple step-stress model with Type-I hybrid censored data from the exponential distribution

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress factors such as temperature, voltage and load on the lifetimes of experimental units. Step-stress test, which is a special class of accelerated life-tests, allows the experimenter to increase the stress levels at fixed times during the experiment in order to obtain information on the parameters of the life distributions more quickly than under normal operating conditions. In this paper, we consider the simple step-stress model under the exponential distribution when the available data are Type-I hybrid censored. We derive the maximum likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model with lifetimes being exponentially distributed. The exact distributions of the MLEs of parameters are obtained through the use of conditional moment generating functions. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs and the parametric bootstrap methods, and assess their performance through a Monte Carlo simulation study. Finally, we present two examples to illustrate all the methods of inference discussed here.