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.     

Parameter estimation for a word frequency distribution based on occupancy theory

Compromise Estimators between the generalized Bayes and Bayes estimators with respect to conjugate gamma priors under entropy loss are proposed. The proposed compromise estimators are compared with some suitable generalized Bayes estimators in terms of their frequentist risk performance. Also the RSL approach will be employed to compare the proposed compromise estimators and some admissible generalized Bayes estimators in terms of their Bayes risk performance.     

Estimation of three-parameter exponentiated-Weibull distribution under type-II censoring

The present article obtains the point estimators of the exponentiated-Weibull parameters when all the three parameters of the distribution are unknown. Maximum likelihood estimator generalized maximum likelihood estimator and Bayes estimators are proposed for three-parameter exponentiated-Weibull distribution when available sample is type-II censored. Independent non-informative types of priors are considered for the unknown parameters to develop generalized maximum likelihood estimator and Bayes estimators. Although the proposed estimators cannot be expressed in nice closed forms, these can be easily obtained through the use of appropriate numerical techniques. The performances of these estimators are studied on the basis of their risks, computed separately under LINEX loss and squared error loss functions through Monte-Carlo simulation technique. An example is also considered to illustrate the estimators.     

ESTIMATION OF EXPONENTIATED WEIBULL SHAPE PARAMETERS UNDER LINEX LOSS FUNCTION

ABSTRACT

The paper deals with Bayes estimation of the exponentiated Weibull shape parameters under linex loss function when independent non-informative type of priors are available for the parameters. Generalized maximum likelihood estimators have also been obtained. Performances of the proposed Bayes estimator, generalized maximum likelihood estimators, posterior mean (i.e., Bayes estimator under squared error loss function) and maximum likelihood estimators have been studied on the basis of their risks under linex loss function. The comparison is based on a simulation study because the expressions for risk functions of these estimators cannot be obtained in nice closed forms.     

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.     

On the Bayesian Estimation for the Uniform Scale Parameter via a Functional Equation

Bayes uniform model under the squared error loss function is shown to be completely identifiable by the form of the Bayes estimates of the scale parameter. This results in solving a specific functional equation. A complete characterization of differentiable Bayes estimators (BE) and generalized Bayes estimators (GBE) is given as well as relations between degrees of smoothness of the estimators and the priors. Characterizations of strong (generalized Bayes) Bayes sequence (SBS or SGBS) are also investigated. A SBS is a sequence of estimators (one for each sample size) where all its components are BE generated by the same prior measure. A complete solution is given for polynomial Bayesian estimation.     

Bayes admissible estimation of the means in Poisson decomposable graphical models

We investigate the Bayes estimation of the means in Poisson decomposable graphical models. Some classes of Bayes estimators are provided which improve on the maximum likelihood estimator under the normalized squared error loss. Both proper and improper priors are included in the proposed classes of priors. Concerning the generalized Bayes estimators with respect to the improper priors, we address their admissibility.     

Bayesian analysis of head and neck cancer data using generalized inverse Lindley stress–strength reliability model

In this article, we present the analysis of head and neck cancer data using generalized inverse Lindley stress–strength reliability model. We propose Bayes estimators for estimating P(X > Y), when X and Y represent survival times of two groups of cancer patients observed under different therapies. The X and Y are assumed to be independent generalized inverse Lindley random variables with common shape parameter. Bayes estimators are obtained under the considerations of symmetric and asymmetric loss functions assuming independent gamma priors. Since posterior becomes complex and does not possess closed form expressions for Bayes estimators, Lindley’s approximation and Markov Chain Monte Carlo techniques are utilized for Bayesian computation. An extensive simulation experiment is carried out to compare the performances of Bayes estimators with the maximum likelihood estimators on the basis of simulated risks. Asymptotic, bootstrap, and Bayesian credible intervals are also computed for the P(X > Y).     

A new class of minimax generalized Bayes estimators of a normal variance

A new class of minimax generalized Bayes estimators of the variance of a normal distribution is given under both quadratic and entropy losses. One contribution of the paper is a new class of minimax generalized Bayes estimators of a particularly simple form. Another contribution is a class of minimax generalized Bayes procedures satisfying a Strawderman [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198]-type condition which do not satisfy a Brewster and Zidek [1974. Improving on equivariant estimators. Ann. Statist. 2, 21–38]-type condition. We indicate that the new class may have a noticeably larger region of substantial improvement over the usual estimator than Brewster and Zidek-type procedures.     

Bayesian estimation for randomly censored generalized exponential distribution under asymmetric loss functions

This paper deals with the Bayesian estimation of generalized exponential distribution in the proportional hazards model of random censorship under asymmetric loss functions. It is well known for the two-parameter lifetime distributions that the continuous conjugate priors for parameters do not exist; we assume independent gamma priors for the scale and the shape parameters. It is observed that the closed-form expressions for the Bayes estimators cannot be obtained; we propose Tierney–Kadane's approximation and Gibbs sampling to approximate the Bayes estimates. Monte Carlo simulation is carried out to observe the behavior of the proposed methods and one real data analysis is performed for illustration. Bayesian methods are compared with maximum likelihood and it is observed that the Bayes estimators perform better than the maximum-likelihood estimators in some cases.     

On the estimation of a normal precision and a normal variance ratio

The problem of estimating the ratio of the variances of two independent normal populations is considered under both quadratic and entropy losses, when the means are unknown. New classes of improved estimators are obtained with the following properties. They are smooth, improve on the risk of the best affine equivariant estimator at every parameter point, have very simple form and are based on all the available data. In the case of entropy loss, the estimators of one of these classes are, additionally, generalized Bayes. Our approach for constructing these improved estimators is based on Strawderman’s (1974) [18] technique. As a preliminary result of independent interest, new classes of dominating generalized Bayes procedures for a normal precision are also given.     

Relative squared error prediction in the generalized linear regression model

In linear regression models, predictors based on least squares or on generalized least squares estimators are usually applied which, however, fail in case of multicollinearity. As an alternative biased estimators like ridge estimators, Kuks-Olman estimators, Bayes or minimax estimators are sometimes suggested. In our analysis the relative instead of the generally used absolute squared error enters the objective function. An explicit minimax solution is derived which, in an important special case, can be viewed as a predictor based on a Kuks-Olman estimator.     

Empirical bayes estimation in contingency tables

In the estimation of cell probabilities from a two–way contingency table, suppose that a priori the classification variables are believed independent. New empirical Bayes and Bayes estimators are proposed which shrink the observed proportions towards classical estimates under the model of independence. The estimators, based on a Dirichlet mixture class of priors, compare favorably to an estimator of Laird (1978) that is based on a normal prior on terms of a log–linear model. The methods are generalized to three–way tables.     

Bayesian estimation of mean and square of mean of normal distribution using linex loss function

In this paper, the Bayes estimators for mean and square of mean ol a normal distribution with mean μ and vaiiance σ r2 (known), relative to LINEX loss function are obtained Comparisons in terms of risk functions and Bayes risks of those under LINEX loss and squared error loss functions with their respective alternative estimators viz, UMVUE and Bayes estimators relative to squared error loss function, are made. It is found that Bayes estimators relative to LINEX loss function dominate the alternative estimators m terms of risk function snd Bayes risk. It is also found that if t2 is unknown the Bayes estimators are still preferable over alternative estimators.     

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.     

Estimation and prediction for the generalized inverted exponential distribution based on progressively first-failure-censored data with application

In this paper, the estimation of parameters for a generalized inverted exponential distribution based on the progressively first-failure type-II right-censored sample is studied. An expectation–maximization (EM) algorithm is developed to obtain maximum likelihood estimates of unknown parameters as well as reliability and hazard functions. Using the missing value principle, the Fisher information matrix has been obtained for constructing asymptotic confidence intervals. An exact interval and an exact confidence region for the parameters are also constructed. Bayesian procedures based on Markov Chain Monte Carlo methods have been developed to approximate the posterior distribution of the parameters of interest and in addition to deduce the corresponding credible intervals. The performances of the maximum likelihood and Bayes estimators are compared in terms of their mean-squared errors through the simulation study. Furthermore, Bayes two-sample point and interval predictors are obtained when the future sample is ordinary order statistics. The squared error, linear-exponential and general entropy loss functions have been considered for obtaining the Bayes estimators and predictors. To illustrate the discussed procedures, a set of real data is analyzed.     

On the Bayes estimators of the parameters of size-biased generalized power series distributions

ABSTRACT

In this paper, we derive the Bayes estimators of functions of parameters of the size-biased generalized power series distribution under squared error loss function and weighted square error loss function. The results of size-biased GPSD are then used to obtain particular cases of the size-biased negative binomial, size-biased logarithmic series, and size-biased Poisson distributions. These estimators are better than the classical minimum variance unbiased estimators in the sense that they increase the range of the estimation. Finally, an example is provided to illustrate the results and a goodness of fit test is done using the maximum likelihood and Bayes estimators.     

Optimal minimax squared error risk estimation of the mean of a multivariate normal distribution

Minimax squared error risk estimators of the mean of a multivariate normal distribution are characterized which have smallest Bayes risk with respect to a spherically symmetric prior distribution for (i) squared error loss, and (ii) zero-one loss depending on whether or not estimates are consistent with the hypothesis that the mean is null. In (i), the optimal estimators are the usual Bayes estimators for prior distributions with special structure. In (ii), preliminary test estimators are optimal. The results are obtained by applying the theory of minimax-Bayes-compromise decision problems.     

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.     

Proper Bayes minimax estimators of the normal mean matrix with common unknown variances

This paper addresses the problem of estimating a matrix of the normal means, where the variances are unknown but common. The approach to this problem is provided by a hierarchical Bayes modeling for which the first stage prior for the means is matrix-variate normal distribution with mean zero matrix and a covariance structure and the second stage prior for the covariance is similar to Jeffreys’ rule. The resulting hierarchical Bayes estimators relative to the quadratic loss function belong to a class of matricial shrinkage estimators. Certain conditions are obtained for admissibility and minimaxity of the hierarchical Bayes estimators.