Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-II censoring

In this paper, the statistical inference of the unknown parameters of a two-parameter inverse Weibull (IW) distribution based on the progressive type-II censored sample has been considered. The maximum likelihood estimators (MLEs) cannot be obtained in explicit forms, hence the approximate MLEs are proposed, which are in explicit forms. The Bayes and generalized Bayes estimators for the IW parameters and the reliability function based on the squared error and Linex loss functions are provided. The Bayes and generalized Bayes estimators cannot be obtained explicitly, hence Lindley's approximation is used to obtain the Bayes and generalized Bayes estimators. Furthermore, the highest posterior density credible intervals of the unknown parameters based on Gibbs sampling technique are computed, and using an optimality criterion the optimal censoring scheme has been suggested. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, two data sets have been analysed for illustrative purposes.     

Empirical bayes estimation of reliability characteristics for an exponential family

This paper considers empirical Bayes (EB) squared-error-loss estimations of mean lifetime, variance and reliability function for failure-time distributions belonging to an exponential family, which includes gamma and Weibull distributions as special cases. EB estimators are proposed when the prior distribution of the lifetime parameter is completely unknown but has a compact (known or unknown) support. Asymptotic optimality and rates of convergence of these estimators are investigated. The rates established here under the compact support restriction are better than the polynomial rates of convergence obtained previously.     

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.     

Bayesian Inference in Dependent Right Censoring

In this article, we consider dependent right censoring when the lifetime and censoring variables have a Marshall–Olkin bivariate exponential distribution and obtain MLEs, MMEs and UMVUEs of the unknown parameters. The Bayes estimators as well as the Posterior Regret Gamma Minimax (PRGM) estimators of the parameters of interest under the SEL function are also obtained and a Monte Carlo simulation study is carried out to compare these estimators.     

Empirical bayes estimation of a distribution function with a gamma process prior

A sequence of empirical Bayes estimators is given for estimating a distribution function. It is shown that ‘i’ this sequence is asymptotically optimum relative to a Gamma process prior, ‘ii’ the overall expected loss approaches the minimum Bayes risk at a rate of n , and ‘iii’ the estimators form a sequence of proper distribution functions. Finally, the numerical example presented by Susarla and Van Ryzin ‘Ann. Statist., 6, 1978’ reworked by Phadia ‘Ann. Statist., 1, 1980, to appear’ has been analyzed and the results are compared to the numerical results by Phadia     

Estimation of the scale parameter of the Rayleigh distribution with multiply type–II censored sample

Based on a multiply type-II censored sample, the maximum likelihood estimator (MLE) and Bayes estimator for the scale parameter and the reliability function of the Rayleigh distribution are derived. However, since the MLE does not exist an explicit form, an approximate MLE which is the maximizer of an approximate likelihood function will be given. The comparisons among estimators are investigated through Monte Carlo simulations. An illustrative example with the real data concerning the 23 ball bearing in the life test is presented.     

Minimax linear empirical bayes estimation of the binomial parameter

The minimax linear Empirical Bayes estimators for a binomial parameter are obtained, assuming some information about the moments of the prior. The form of these estimates is used to propose a criterion which may be helpful in determining whether Empirical Bayes estimation is Indicated for a given problem.     

A New Approach to the Estimation of Inter-Variable Correlation

The use of different measures of similarity between observed vectors for the purposes of classifying or clustering them has been expanding dramatically in recent years. One result of this expansion has been the use of many new similarity measures, designed for the purpose of satisfying various criteria. A noteworthy application involves estimating the relationships between genes using microarray experimental data. We consider the class of ‘correlation-type’ similarity measures. The use of these new measures of similarity suggest that the whole problem needs to be formulated in statistical terms to clarify their relative benefits. Pursuant to this need, we define, for each given observed vector, a baseline representing the ‘true’ value common to each of the component observations. These ‘true’ values are taken to be parameters. We define the ‘true correlation’ between each two observed vectors as the average (over the distribution of the observations for given baseline parameters) of Pearson's correlation with sample means replaced by the corresponding baseline parameters. Estimators of this true correlation are assessed using their mean squared error (MSE). Proper Bayes estimators of this true correlation, being based on the predictive posterior distribution of the data, are both difficult to calculate/analyze and highly non robust. By constrast, empirical Bayes estimators are: (i) close to their Bayesian counterparts; (ii) easy to analyze; and (iii) strongly robust. For these reasons, we employ empirical Bayes estimators of correlation in place of their Bayesian counterparts. We show how to construct two different kinds of simultaneous Bayes correlation estimators: the first assumes no apriori correlation between baseline parameters; the second assumes a common unknown correlation between them. Estimators of the latter type frequently have significantly smaller MSE than those of the former type which, in turn, frequently have significantly smaller MSE than their Pearson estimator counterparts. For purposes of illustrating our results, we examine the problem of inferring the relationships between gene expression level vectors, in the context of observing microarray experimental data.     

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.     

Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring

We consider the problem of estimating unknown parameters, reliability function and hazard function of a two parameter bathtub-shaped distribution on the basis of progressive type-II censored sample. The maximum likelihood estimators and Bayes estimators are derived for two unknown parameters, reliability function and hazard function. The Bayes estimators are obtained against squared error, LINEX and entropy loss functions. Also, using the Lindley approximation method we have obtained approximate Bayes estimators against these loss functions. Some numerical comparisons are made among various proposed estimators in terms of their mean square error values and some specific recommendations are given. Finally, two data sets are analyzed to illustrate the proposed methods.     

Classical and Bayesian estimation of P(Y < X) for Kumaraswamy's distribution

The aim of this paper is to study the estimation of the reliability R=P(Y<X) when X and Y are independent random variables that follow Kumaraswamy's distribution with different parameters. If we assume that the first shape parameter is common and known, the maximum-likelihood estimator (MLE), the exact confidence interval and the uniformly minimum variance unbiased estimator of R are obtained. Moreover, when the first parameter is common but unknown, MLEs, Bayes estimators, asymptotic distributions and confidence intervals for R are derived. Furthermore, Bayes and empirical Bayes estimators for R are obtained when the first parameter is common and known. Finally, when all four parameters are different and unknown, the MLE of R is obtained. Monte Carlo simulations are performed to compare the different proposed methods and conclusions on the findings are given.     

Bayes and empirical bayes estimation of the probability that z > x + y

Let X, Y and Z be independent random variables with common unknown distribution F. Using the Dirichlet process prior for F and squared erro loss function, the Bayes and empirical Bayes estimators of the parameters λ(F). the probability that Z > X + Y, are derived. The limiting Bayes estimator of λ(F) under some conditions on the parameter of the process is shown to be asymptotically normal. The aysmptotic optimality of the empirical Bayes estimator of λ(F) is established. When X, Y and Z have support on the positive real line, these results are derived for randomly right censored data. This problem relates to testing whether than used discussed by Hollander and Proshcan (1972) and Chen, Hollander and Langberg (1983).     

Bayesian estimation based on progressive Type-II censoring from two-parameter bathtub-shaped lifetime model: an Markov chain Monte Carlo approach

In this paper, maximum likelihood and Bayes estimators of the parameters, reliability and hazard functions have been obtained for two-parameter bathtub-shaped lifetime distribution when sample is available from progressive Type-II censoring scheme. The Markov chain Monte Carlo (MCMC) method is used to compute the Bayes estimates of the model parameters. It has been assumed that the parameters have gamma priors and they are independently distributed. Gibbs within the Metropolis–Hasting algorithm has been applied to generate MCMC samples from the posterior density function. Based on the generated samples, the Bayes estimates and highest posterior density credible intervals of the unknown parameters as well as reliability and hazard functions have been computed. The results of Bayes estimators are obtained under both the balanced-squared error loss and balanced linear-exponential (BLINEX) loss. Moreover, based on the asymptotic normality of the maximum likelihood estimators the approximate confidence intervals (CIs) are obtained. In order to construct the asymptotic CI of the reliability and hazard functions, we need to find the variance of them, which are approximated by delta and Bootstrap methods. Two real data sets have been analyzed to demonstrate how the proposed methods can be used in practice.     

Bayes estimation of reliability for the two-parameter lognormal distribution

Bayes estimators of reliability for the lognormal failure distribution with two parameters (M,∑) are obtained both for informative priors of normal-gamma type and for the vague prior of Jeffreys. The estimators are in terms of the t-distribution function. The Bayes estimators are compared with the maximum likelihood and minimum variance unbiased estimators of reliabil-ity using Monte Carlo simulations.     

Bayes estimators in the presence of a guess value

The paper deals with the problem of parameter estimation in the presence of a guess value and attempts to justify the use of Bayes estimators as an alternative to ordinary shrinkage estimators. Finally, certain Bayes estimators of exponential parameters are obtained under type II censoring, and these are compared with the corresponding MLEs and ordinary shrinkage estimators using a Monte Carlo study.     

Use of approximate bayesian methods for the block and basu bivariate exponential distribution

Summary In this paper, we present a Bayesian analysis of the bivariate exponential distribution of Block and Basu (1974) assuming different prior densities for the parameters of the model and considering Laplace's method to obtain approximate marginal posterior and posterior moments of interest. We also find approximate Bayes estimators for the reliability of two-component systems at a specified timet ₀ considering series and parallel systems. We illustrate the proposed methodology with a generated data set.     

Estimation of parameters of inverse Weibull distribution and application to multi-component stress-strength model

The problem of estimation of the parameters of two-parameter inverse Weibull distributions has been considered. We establish existence and uniqueness of the maximum likelihood estimators of the scale and shape parameters. We derive Bayes estimators of the parameters under the entropy loss function. Hierarchical Bayes estimator, equivariant estimator and a class of minimax estimators are derived when shape parameter is known. Ordered Bayes estimators using information about second population are also derived. We investigate the reliability of multi-component stress-strength model using classical and Bayesian approaches. Risk comparison of the classical and Bayes estimators is done using Monte Carlo simulations. Applications of the proposed estimators are shown using real data sets.     

Dependent Right Censorship in the MOMW Distribution

In this paper, we consider paired survival data, in which pair members are subject to the same right censoring time, but they are dependent on each other. Assuming the Marshall–Olkin Multivariate Weibull distribution for the joint distribution of the lifetimes (X₁, X₂) and the censoring time X₃, we derive the joint density of the actual observed data and obtain maximum likelihood estimators, Bayes estimators and posterior regret Gamma minimax estimators of the unknown parameters under squared error loss and weighted squared error loss functions. We compare the performances of the maximum likelihood estimators and Bayes estimators numerically in terms of biases and estimated Mean Squared Error Loss.     

Extended linear empirical bayes estimation

In this paper, the linear empirical Bayes estimation method, which is based on approximation of the Bayes estimator by a linear function, is generalized to an extended linear empirical Bayes estimation technique which represents the Bayes estimator by a series of algebraic polynomials. The extended linear empirical Bayes estimators are elaborated in the case of a location or a scale parameter. The theory is illustrated by examples of its application to the normal distribution with a location parameter and the gamma distribution with a scale parameter. The linear and the extended linear empirical Bayes estimators are constructed in these two cases and, then, studied numerically via Monte Carlo simulations. The simulations show that the extended linear empirical Bayes estimators have better convergence rates than the traditional linear empirical Bayes estimators.     

Bayes methods in the ecological fallacy cntext:estimation of individual correlation from aggregate data

The ecological fallacy is related to Simpson's paradox (1951) where relationships among group means may be counterintuitive and substantially different from relationships within groups, where the groups are usually geographic entities such as census tracts. We consider the problem of estimating the correlation between two jointly normal random variables where only ecological data (group means) are available. Two empirical Bayes estimators and one fully Bayesian estimator are derived and compared with the usual ecological estimator, which is simply the Pearson correlation coefficient of the group sample means. We simulate the bias and mean squared error performance of these estimators, and also give an example employing a dataset where the individual level data are available for model checking. The results indicate superiority of the empirical Bayes estimators in a variety of practical situations where, though we lack individual level data, other relevant prior information is available.