Comparison of linex and quadratic bayes estimators foe the rayleigh distribution

In this paper, the Bayes estimators for the parameter, the reliability function, and failure rate function of the Rayleigh distribution are obtained when based on complete or type II censored samples. Some types of the linex loss function are used. Comparieons in terms of risks of those under linex loss and squared error loss function with Bayes estimators relative to squared error loss function are made, Numerical example and simulation example are included.     

Bayesian Estimation for Poisson-exponential Model under Progressive Type-II Censoring Data with Binomial Removal and Its Application to Ovarian Cancer Data

In this article, we propose Maximum likelihood estimators (MLEs) and Bayes estimators of parameters of Poisson-exponential distribution (PED) under General entropy loss function (GELF) and Squared error loss function (SELF) for Progressive type-II censored data with binomial removals (PT-II CBRs). The MLEs and corresponding Bayes estimators are compared in terms of their risks based on simulated samples from PED. The proposed methodology is illustrated on a real dataset of ovarian cancer.     

The Bayes rule of the parameter in (0,1) under Zhang’s loss function with an application to the beta-binomial model

Abstract

For the restricted parameter space (0,1), we propose Zhang’s loss function which satisfies all the 7 properties for a good loss function on (0,1). We then calculate the Bayes rule (estimator), the posterior expectation, the integrated risk, and the Bayes risk of the parameter in (0,1) under Zhang’s loss function. We also calculate the usual Bayes estimator under the squared error loss function, and the Bayes estimator has been proved to underestimate the Bayes estimator under Zhang’s loss function. Finally, the numerical simulations and a real data example of some monthly magazine exposure data exemplify our theoretical studies of two size relationships about the Bayes estimators and the Posterior Expected Zhang’s Losses (PEZLs).     

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.     

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.     

On the Bayesian analysis of the mixture of power function distribution using the complete and the censored sample

The power function distribution is often used to study the electrical component reliability. In this paper, we model a heterogeneous population using the two-component mixture of the power function distribution. A comprehensive simulation scheme including a large number of parameter points is followed to highlight the properties and behavior of the estimates in terms of sample size, censoring rate, parameters size and the proportion of the components of the mixture. The parameters of the power function mixture are estimated and compared using the Bayes estimates. A simulated mixture data with censored observations is generated by probabilistic mixing for the computational purposes. Elegant closed form expressions for the Bayes estimators and their variances are derived for the censored sample as well as for the complete sample. Some interesting comparison and properties of the estimates are observed and presented. The system of three non-linear equations, required to be solved iteratively for the computations of maximum likelihood (ML) estimates, is derived. The complete sample expressions for the ML estimates and for their variances are also given. The components of the information matrix are constructed as well. Uninformative as well as informative priors are assumed for the derivation of the Bayes estimators. A real-life mixture data example has also been discussed. The posterior predictive distribution with the informative Gamma prior is derived, and the equations required to find the lower and upper limits of the predictive intervals are constructed. The Bayes estimates are evaluated under the squared error loss function.     

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.     

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.     

Reliability Characteristics of the Maxwell Distribution: A Bayes Estimation Study

ABSTRACT

This article presents maximum likelihood, Bayes, and empirical Bayes estimators of the truncated first moment and hazard function of the Maxwell distribution. A comparison of the relative efficiency of these three estimators is performed via a Monte Carlo simulation study.     

The empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior under Stein's loss function

For the hierarchical Poisson and gamma model, we calculate the Bayes posterior estimator of the parameter of the Poisson distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior by two methods. In numerical simulations, we have illustrated: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we exploit the attendance data on 314 high school juniors from two urban high schools to illustrate our theoretical studies.     

Bayesian and Non Bayesian Estimations on the Exponentiated Modified Weibull Distribution for Progressive Censored Sample

The four-parameter Exponentiated Modified Weibull (EMW) is considered as an important lifetime distribution. Based on progressive Type-II censored sample, maximum likelihood and Bayesian estimators of the parameters, reliability function, and hazard rate function are derived. Two cases are considered: first, the case of one unknown exponent parameter of EMW and second, the case when two parameters of the EMW are both unknown. The Bayes estimators are studied under squared error and LINEX loss functions. The standard Bayes and importance sampling are considered for the estimation. Monte Carlo simulations are performed under different samples sizes and different censoring schemes for investigating and comparing the methods of estimation.     

Inference based on progressively censored sample from Pareto population

Abstract

In this paper, we assume that the lifetimes have a two-parameter Pareto distribution and discuss some results of progressive Type-II censored sample. We obtain maximum likelihood estimators and Bayes estimators of the unknown parameters under squared error loss and a precautionary loss functions in progressively Type-II censored sample. Robust Bayes estimation of unknown parameters over three different classes of priors under progressively Type-II censored sample, squared error loss, and precautionary loss functions are obtained. We discuss estimation of unknown parameters on competing risks progressive Type-II censoring. Finally, we consider the problem of estimating the common scale parameter of two Pareto distributions when samples are progressively Type-II censored.     

Estimation of population distribution function involving measurement error in the presence of non response

Abstract

This article addresses the problem of estimating population distribution function for simple random sampling in the presence of non response and measurement error together. We suggest a general class of estimators for estimating the cumulative distribution function using the auxiliary information. The expressions for the bias and mean squared error are derived up to the first order of approximation. The performance of the proposed class of estimators is compared with considered estimators both theoretically and numerically. A real data set is used to support the theoretical findings.     

Loss functions in restricted parameter spaces and their Bayesian applications

ABSTRACT

Squared error loss remains the most commonly used loss function for constructing a Bayes estimator of the parameter of interest. However, it can lead to suboptimal solutions when a parameter is defined on a restricted space. It can also be an inappropriate choice in the context when an extreme overestimation and/or underestimation results in severe consequences and a more conservative estimator is preferred. We advocate a class of loss functions for parameters defined on restricted spaces which infinitely penalize boundary decisions like the squared error loss does on the real line. We also recall several properties of loss functions such as symmetry, convexity and invariance. We propose generalizations of the squared error loss function for parameters defined on the positive real line and on an interval. We provide explicit solutions for corresponding Bayes estimators and discuss multivariate extensions. Four well-known Bayesian estimation problems are used to demonstrate inferential benefits the novel Bayes estimators can provide in the context of restricted estimation.     

BAYES EMPIRICAL BAYES APPROACH TO ESTIMATION OF THE FAILURE RATE IN EXPONENTIAL DISTRIBUTION

ABSTRACT

In the empirical Bayes (EB) decision problem consisting of squared error estimation of the failure rate in exponential distribution, a prior Λ is placed on the gamma family of prior distributions to produce Bayes EB estimators which are admissible. A subclass of such estimators is shown to be asymptotically optimal (a.o.). The results of a Monte Carlo study are presented to demonstrate the a.o. property of the Bayes EB estimators.     

Bayesian estimation for non zero inflated modified power series distribution under linex and generalized entropy loss functions

Abstract

In this paper, we derive Bayesian estimators of the parameters of modified power series distributions inflated at any of a support point under linex and general entropy loss function. We assume that the prior information can be summarized by a uniform, Beta, two-sided power, Gamma or generalized Pareto distributions. The obtained results are demonstrated on the generalized Poisson and the generalized negative binomial distribution inflated at a given point.     

Bayes Estimator of Inverse Gaussian Parameters Under General Entropy Loss Function Using Lindley's Approximation

In this article, we obtained Bayes estimators of parameters of Inverse Gaussian distributions under asymmetric loss function using Lindley's Approximation (L-Approximation). The proposed estimators have been compared with the corresponding estimators obtained under symmetric loss function and MLE for their risks. This comparison is illustrated using Monte-Carlo study of 2,000 simulated sample from the Inverse Gaussian distribution.     

Bayesian inference of three-parameter bathtub-shaped lifetime distribution

We investigate a Bayesian inference in the three-parameter bathtub-shaped lifetime distribution which is obtained by adding a power parameter to the two-parameter bathtub-shaped lifetime distribution suggested by Chen (2000). The Bayes estimators under the balanced squared error loss function are derived for three parameters. Then, we have used Lindley's and Tierney–Kadane approximations (see Lindley 1980; Tierney and Kadane 1986) for computing these Bayes estimators. In particular, we propose the explicit form of Lindley's approximation for the model with three parameters. We also give applications with a simulated data set and two real data sets to show the use of discussed computing methods. Finally, concluding remarks are mentioned.     

Minimaxity of empirical bayes estimators of the means of independent normal variables with unequal variances

The problem of simultaneous estimation of normal means is considered when variances are unequal and the loss is sum of squared errors. Minimaxity or non-minimaxity of empirical Bayes estimators is investigated when the common prior distribution is given by normal one with mean 0. Minimaxity results for the case when the loss is a weighted sum of squared errors is also given. Monte Carlo simulation results are given to compare the risk behavior of the empirical Bayes estimator with those of other minimax ones.