Nonparametric bayes estimation of the survival function from a record of failures and follow-ups

In some observational studies, we have random censoring model. However, the data available may be partially observable censored data consisting of the observed failure times and only those nonfailure times which are subject to follow-up. Suzuki (1985) discussed the problem of nonparametric estimation of the survival function from such partially observable censored data. In this article, we derive a nonparametric Bayes estimator of the survival function for such data of failures and follow-ups under a Dirichlet process prior and squared error loss. The limiting properties such as the mean square consistency, weak convergence and strong consistency of the Bayes estimator are studied. Finally, the procedures developed are illustrated by means of an example.     

Bayes prediction of number of failures in weibull samples

This work is concerned with the Bayesian prediction problem of the number of components which will fail in a future time interval, when the failure times are Weibull distributed. Both the 1-sample and the 2-sample prediction problems are dealed with, and some choices of the prior densities on the distribution parameters are discussed which are relatively easy to work with and allow different degrees of knowledge on the failure mechanism to be incorporated in the predictive procedure. Useful relations between the predictive distribution on the number of future failures and the predictive distribution on the future failure times are derived. Numerical examples are also given.     

Empirical Bayes estimation in continuous one-parameter exponential families under associated samples

In this paper, we study the empirical Bayes (EB) estimation in continuous one-parameter exponential families under negatively associated (NA) samples and positively associated (PA) samples. Under certain regularity conditions, it is shown that the convergence rates of proposed EB estimators under NA or PA samples are the same as those of EB estimators under independent observations, which significantly improve the existing results in EB estimation under associated samples.     

Hierarchical bayes quality measurement plan

Quality Measurement Plan (QMP) as developed by Hoadley (1981) is a statistical method for analyzing discrete quality audit data which consist of the expected number of defects given the standard quality. The QMP is based on an empirical Bayes (EB) model of the audit sampling process. Despite its wide publicity, Hoadley's method has often been described as heuristic. In this paper we offer an hierarchical Bayes (HB) alternative to Hoadley's EB model, and overcome much of the criticism against this model. Gibbs sampling is used to implement the HB model proposed in this paper. Also, the convergence of the Gibbs sampler is monitored via the algorithm of Gelman and Rubin (1992).     

A generalization of the compound rayleigh distribution: using a bayesian method on cancer survival times

In this paper the generalized compound Rayleigh model, exhibiting flexible hazard rate, is high¬lighted. This makes it attractive for modelling survival times of patients showing characteristics of a random hazard rate. The Bayes estimators are derived for the parameters of this model and some survival time parameters from a right censored sample. This is done with respect to conjugate and discrete priors on the parameters of this model, under the squared error loss function, Varian's asymmetric linear-exponential (linex) loss function and a weighted linex loss function. The future survival time of a patient is estimated under these loss functions. A Monte Carlo simu¬lation procedure is used where closed form expressions of the estimators cannot be obtained. An example illustrates the proposed estimators for this model.     

Bayes Analysis and Comparison of Accelerated Weibull and Accelerated Birnbaum–Saunders Models

Several models are proposed in the literature for modeling fatigue data resulting from materials subject to cyclic stress and strain. Accelerated Weibull and accelerated Birnbaum–Saunders distributions are most commonly used models. Whereas the accelerated Weibull model is easier compared to accelerated Birnbaum–Saunders, it fails to represent the situation equally well. The present article focuses on Bayes analysis of the two models and provides a comparison based on some important Bayesian tools. Model compatibility study using predictive simulation ideas is preceded by the said comparison. Throughout, the posterior simulations are carried out by Markov chain Monte Carlo procedure.     

Bayesian ratemaking under Dirichlet process mixtures

Experience ratemaking plays a crucial role in general insurance in determining future premiums of individuals in a portfolio by assessing observed claims from the whole portfolio. This paper investigates this problem in which claims can be modeled by certain parametric family of distributions. The Dirichlet process mixtures are employed to model the distributions of the parameters so as to make two advantages: to produce exact Bayesian experience premiums for a class of premium principles generated from generic error functions and, at the same time, provide robust and flexible ways to avoid possible bias caused by traditionally used priors such as non informative priors or conjugate priors. In this paper, the Bayesian experience ratemaking under Dirichlet process mixture models are investigated and due to the lack of analytical forms of the conditional expectations of the quantities concerned, the Gibbs sampling schemes are designed for the purpose of approximations.     

Confidence distributions applied to propagating uncertainty to inference based on estimating the local false discovery rate: A fiducial continuum from confidence sets to empirical Bayes set estimates as the number of comparisons increases

Empirical Bayes estimates of the local false discovery rate can reflect uncertainty about the estimated prior by supplementing their Bayesian posterior probabilities with confidence levels as posterior probabilities. This use of coherent fiducial inference with hierarchical models generates set estimators that propagate uncertainty to varying degrees. Some of the set estimates approach estimates from plug-in empirical Bayes methods for high numbers of comparisons and can come close to the usual confidence sets given a sufficiently low number of comparisons.     

Estimation under modified Weibull distribution based on right censored generalized order statistics

In this paper, the maximum likelihood (ML) and Bayes, by using Markov chain Monte Carlo (MCMC), methods are considered to estimate the parameters of three-parameter modified Weibull distribution (MWD(β, τ, λ)) based on a right censored sample of generalized order statistics (gos). Simulation experiments are conducted to demonstrate the efficiency of the proposed methods. Some comparisons are carried out between the ML and Bayes methods by computing the mean squared errors (MSEs), Akaike's information criteria (AIC) and Bayesian information criteria (BIC) of the estimates to illustrate the paper. Three real data sets from Weibull(α, β) distribution are introduced and analyzed using the MWD(β, τ, λ) and also using the Weibull(α, β) distribution. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic, {AIC and BIC} to emphasize that the MWD(β, τ, λ) fits the data better than the other distribution. All parameters are estimated based on type-II censored sample, censored upper record values and progressively type-II censored sample which are generated from the real data sets.     

The umvue and bayes estimate of reliability of mixed failure time distribution

We study the reliability estimates of the non-standard mixture of degenerate (degenerated at zero) and exponential distributions. The Uniformly Minimum Variance Unbiased Estimator (UMVUE) and Bayes estimator of the reliability for some selective prior when the mixing proportion is known and unknown are derived. The Bayes risk is computed for each Bayes estimator of the reliability. A simulated study is carried out to assess the performance of the estimators alongwith the true and Maximum Likelihood Estimate (MLE) of the reliability. An example from Vannman (1991) is also discussed at the end of the paper.