Point predictors of the latent failure times of censored units in progressively Type-II censored competing risks data from the exponential distributions

In this paper, we discuss the problem of predicting times to the latent failures of units censored in multiple stages in a progressively Type-II censored competing risks model. It is assumed that the lifetime distribution of the latent failure times are independent and exponential-distributed with the different scale parameters. Several classical point predictors such as the maximum likelihood predictor, the best unbiased predictor, the best linear unbiased predictor, the median unbiased predictor and the conditional median predictor are obtained. The Bayesian point predictors are derived under squared error loss criterion. Moreover, the point estimators of the unknown parameters are obtained using the observed data and different point predictors of the latent failure times. Finally, Monte-Carlo simulations are carried out to compare the performances of the different methods of prediction and estimation and one real data is used to illustrate the proposed procedures.     

Posterior analysis of the compound Rayleigh distribution under balanced loss functions for censored data

This paper develops Bayesian analysis in the context of progressively Type II censored data from the compound Rayleigh distribution. The maximum likelihood and Bayes estimates along with associated posterior risks are derived for reliability performances under balanced loss functions by assuming continuous priors for parameters of the distribution. A practical example is used to illustrate the estimation methods. A simulation study has been carried out to compare the performance of estimates. The study indicates that Bayesian estimation should be preferred over maximum likelihood estimation. In Bayesian estimation, the balance general entropy loss function can be effectively employed for optimal decision-making.     

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.     

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.     

Approximate Bayesian analysis of doubly censored samples from mixture of two Weibull distributions

The purpose of the paper is to estimate the parameters of the two-component mixture of Weibull distribution under doubly censored samples using Bayesian approach. The choice of Weibull distribution is made due to its (i) capability to model failure time data from engineering, medical and biological sciences (ii) added advantages over the well-known lifetime distributions such as exponential, Raleigh, lognormal and gamma distribution in terms of flexibility, increasing and decreasing hazard rate and closed-form distribution function and hazard rate. The proposed two-component mixture of Weibull distribution is even more flexible than its conventional form. However, the estimation of the parameters from the proposed mixture is more complex. Further, we have assumed couple of loss functions under non informative prior for the Bayesian analysis of the parameters from the mixture model. As the resultant Bayes estimators and associated posterior risks cannot be derived in the closed form, we have used the importance sampling and Lindley’s approximation to obtain the approximate estimates for the parameters of the mixture model. The comparison between the performances of approximation techniques has been made on the basis of simulation study and real-life data analysis. The importance sampling is found to be better than Lindley’s approximation as it gives better estimation for shape and mixing parameters of the mixture model and computations under this technique are much easier/shorter than those under Lindley’s approximation.     

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.     

Test for independence between time to failure and cause of failure in competing risks with k causes

In this paper, we develop a simple nonparametric test for testing the independence of time to failure and cause of failure in competing risks set up. We generalise the test to the situation where failure data is right censored. We obtain the asymptotic distribution of the test statistics for complete and censored data. The efficiency loss due to censoring is studied using Pitman efficiency. The performance of the proposed test is evaluated through simulations. Finally we illustrate our test procedure using three real data sets.     

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.     

Reinforced urns and the subdistribution beta‐Stacy process prior for competing risks analysis

In this paper, we introduce the subdistribution beta‐Stacy process, a novel Bayesian nonparametric process prior for subdistribution functions useful for the analysis of competing risks data. In particular, we (i) characterize this process from a predictive perspective by means of an urn model with reinforcement, (ii) show that it is conjugate with respect to right‐censored data, and (iii) highlight its relations with other prior processes for competing risks data. Additionally, we consider the subdistribution beta‐Stacy process prior in a nonparametric regression model for competing risks data, which, contrary to most others available in the literature, is not based on the proportional hazards assumption.     

Bayesian inference based on a jointly type-II censored sample from two exponential populations

In this paper, based on a jointly type-II censored sample from two exponential populations, the Bayesian inference for the two unknown parameters are developed with the use of squared-error, linear-exponential and general entropy loss functions. The problem of predicting the future failure times, both point and interval prediction, based on the observed joint type-II censored data, is also addressed from a Bayesian viewpoint. A Monte Carlo simulation study is conducted to compare the Bayesian estimators with the maximum likelihood estimator developed by Balakrishnan and Rasouli [Exact likelihood inference for two exponential populations under joint type-II censoring. Comput Stat Data Anal. 2008;52:2725–2738]. Finally, a numerical example is utilized for the purpose of illustration.     

Bayesian prediction based on Pareto doubly censored data

Based on Doubly type II censored data, this paper present Bayesian prediction intervals for future ordered failure times of components whose failure times have the classical Pareto distribution. Two different sampling schemes have been considered. Conjugate priors for either the one or the two-parameter cases are outlined. Illustrative examples and a simulation study are included.     

Analysis of Type-II hybrid censored competing risks data

Kundu and Gupta [Analysis of hybrid life-tests in presence of competing risks. Metrica. 2007;65:159–170] provided the analysis of Type-I hybrid censored competing risks data, when the lifetime distributions of the competing cause of failures follows exponential distribution. In this paper, we consider the analysis of Type-II hybrid censored competing risks data. It is assumed that latent lifetime distributions of the competing causes of failures follow independent exponential distributions with different scale parameters. It is observed that the maximum likelihood estimators of the unknown parameters do not always exist. We propose the modified estimators of the scale parameters, which coincide with the corresponding maximum likelihood estimators when they exist, and asymptotically they are equivalent. We obtain the exact distribution of the proposed estimators. Using the exact distributions of the proposed estimators, associated confidence intervals are obtained. The asymptotic and bootstrap confidence intervals of the unknown parameters are also provided. Further, Bayesian inference of some unknown parametric functions under a very flexible Beta-Gamma prior is considered. Bayes estimators and associated credible intervals of the unknown parameters are obtained using the Monte Carlo method. Extensive Monte Carlo simulations are performed to see the effectiveness of the proposed estimators and one real data set has been analysed for the illustrative purposes. It is observed that the proposed model and the method work quite well for this data set.     

Bayesian analysis of censored linear regression models with scale mixtures of normal distributions

As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyzes might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Student-t, Pearson type VII, slash and the contaminated normal distributions, as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. The likelihood function is utilized to compute not only some Bayesian model selection measures but also to develop Bayesian case-deletion influence diagnostics based on the q-divergence measure. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with applications using real and simulated data.     

The Robustness of Several Estimators of the Survivorship Function with Randomly Censored Data

The problem of estimating the survivorship function, R(t) = P(T > t), arises frequently in both engineering and biomedical sciences. In many applications the data one sees are censored due to the occurrence of some competing cause of failure such as withdrawal from the study, failure from some cause not under study, etc. In the biomedical sciences the distribution free estimator suggested by Kaplan and Meier (JASA 1958) is routinely used, while in the engineering sciences a parametric approach is more commonly used. In this report we study the efficiency of these two techniques when a particular parametric model such as the exponential, Weibull, normal, log normal, exponential power, Pareto, Gompertz, gamma, or bathtub shaped hazard distribution is assumed under a variety of censoring schemes and underlying failure models. We conclude that in most cases the parametric estimators outperform the distribution free estimator. The results are particularly striking if the Weibull forms of these estimators are used routinely.     

On progressively censored generalized inverted exponential distribution

A generalized version of inverted exponential distribution (IED) is considered in this paper. This lifetime distribution is capable of modeling various shapes of failure rates, and hence various shapes of aging criteria. The model can be considered as another useful two-parameter generalization of the IED. Maximum likelihood and Bayes estimates for two parameters of the generalized inverted exponential distribution (GIED) are obtained on the basis of a progressively type-II censored sample. We also showed the existence, uniqueness and finiteness of the maximum likelihood estimates of the parameters of GIED based on progressively type-II censored data. Bayesian estimates are obtained using squared error loss function. These Bayesian estimates are evaluated by applying the Lindley's approximation method and via importance sampling technique. The importance sampling technique is used to compute the Bayes estimates and the associated credible intervals. We further consider the Bayes prediction problem based on the observed samples, and provide the appropriate predictive intervals. Monte Carlo simulations are performed to compare the performances of the proposed methods and a data set has been analyzed for illustrative purposes.     

Analysis of progressively type-I interval censored competing risks data for a class of an exponential distribution

ABSTRACT

In this paper, we consider some problems of point estimation and point prediction when the competing risks data from a class of exponential distribution are progressive type-I interval censored. The maximum likelihood estimation and mid-point approximation method are proposed for the estimations of parameters. Also several point predictors of censored units such as the maximum likelihood predictor, the best unbiased predictor and the conditional median predictor are obtained. The methods discussed here are applied when the lifetime distributions of the latent failure times are independent and Weibull-distributed. Finally a simulation study is given by using Monte-Carlo simulations to compare the performances of the different methods and one data analysis has been presented for illustrative purposes.     

Bayesian analysis of bivariate competing risks models with covariates

Bivariate exponential models have often been used for the analysis of competing risks data involving two correlated risk components. Competing risks data consist only of the time to failure and cause of failure. In situations where there is positive probability of simultaneous failure, possibly the most widely used model is the Marshall–Olkin (J. Amer. Statist. Assoc. 62 (1967) 30) bivariate lifetime model. This distribution is not absolutely continuous as it involves a singularity component. However, the likelihood function based on the competing risks data is then identifiable, and any inference, Bayesian or frequentist, can be carried out in a straightforward manner. For the analysis of absolutely continuous bivariate exponential models, standard approaches often run into difficulty due to the lack of a fully identifiable likelihood (Basu and Ghosh; Commun. Statist. Theory Methods 9 (1980) 1515). To overcome the nonidentifiability, the usual frequentist approach is based on an integrated likelihood. Such an approach is implicit in Wada et al. (Calcutta Statist. Assoc. Bull. 46 (1996) 197) who proved some related asymptotic results. We offer in this paper an alternative Bayesian approach. Since systematic prior elicitation is often difficult, the present study focuses on Bayesian analysis with noninformative priors. It turns out that with an appropriate reparameterization, standard noninformative priors such as Jeffreys’ prior and its variants can be applied directly even though the likelihood is not fully identifiable. Two noninformative priors are developed that consist of Laplace's prior for nonidentifiable parameters and Laplace's and Jeffreys's priors for identifiable parameters. The resulting Bayesian procedures possess some frequentist optimality properties as well. Finally, these Bayesian methods are illustrated with analyses of a data set originating out of a lung cancer clinical trial conducted by the Eastern Cooperative Oncology Group.     

Estimation on dependent right censoring scheme in an ordinary bivariate geometric distribution

Discrete lifetime data are very common in engineering and medical researches. In many cases the lifetime is censored at a random or predetermined time and we do not know the complete survival time. There are many situations that the lifetime variable could be dependent on the time of censoring. In this paper we propose the dependent right censoring scheme in discrete setup when the lifetime and censoring variables have a bivariate geometric distribution. We obtain the maximum likelihood estimators of the unknown parameters with their risks in closed forms. The Bayes estimators as well as the constrained Bayes estimates of the unknown parameters under the squared error loss function are also obtained. We considered an extension to the case where covariates are present along with the data. Finally we provided a simulation study and an illustrative example with a real data.     

Prediction for future failures in Weibull distribution under hybrid censoring

In this paper, we consider the prediction of a future observation based on a type-I hybrid censored sample when the lifetime distribution of experimental units is assumed to be a Weibull random variable. Different classical and Bayesian point predictors are obtained. Bayesian predictors are obtained using squared error and linear-exponential loss functions. We also provide a simulation consistent method for computing Bayesian prediction intervals. Monte Carlo simulations are performed to compare the performances of the different methods, and one data analysis has been presented for illustrative purposes.     

The Correct “Ball Bearings” Data

The famous data on fatigue failure times of ball bearings have been quoted incorrectly from Lieblein and Zelen's original paper. The correct data include censored values, as well as non-fatigue failures that must be handled appropriately. They could be described by a mixture of Weibull distributions, corresponding to different modes of failure.