Bayes Prediction for a Heteroscedastic Regression Superpopulation Model Using Balanced Loss Function

We consider Prais–Houthakker heteroscedastic normal regression model having variance of the dependent variable same as square of its expectation. Bayes predictors for the regression coefficient and the mean of a finite population are derived using Zellner's balanced loss function. Bayes predictive expected losses are obtained and compared with those of classical predictors and Bayes predictors under squared error loss function to examine their loss robustness.     

Bayes Predictor of One-Parameter Exponential Family Type Population Mean Under Balanced Loss Function

We obtain a Bayes predictor and a Bayes prediction risk of the mean of a finite population relative to the balanced loss function. The predictive expected losses associated with classical and standard Bayes predictors are derived and compared with that of a Bayes predictor under a balanced loss function. Specific expressions for a regular exponential family distributed superpopulation are presented and illustrated for some well-known superpopulations.     

Bayes Estimation Based on Random Censored Data for Some Life Time Models Under Symmetric and Asymmetric Loss Functions

Censored data arise naturally in a number of fields, particularly in problems of reliability and survival analysis. There are several types of censoring; in this article, we shall confine ourselves to the right randomly censoring type. Under the Bayesian framework, we study the estimation of parameters in a general framework based on the random censored observations under Linear-Exponential (LINEX) and squared error loss (SEL) functions. As a special case, Weibull model is discussed and the admissibility of estimators of parameters verified. Finally, a simulation study is conducted based on Monte Carlo (MC) method for comparing estimated risks of the estimators obtained.     

On Bayes Linear Unbiased Estimator Under the Balanced Loss Function

In this paper, the Bayes linear unbiased estimator (Bayes LUE) is derived under the balanced loss function. Moreover, the superiority of Bayes LUE over ordinary least square estimator is studied under the mean square error matrix criterion and Pitman closeness criterion. Furthermore, we compare Bayes LUE under the balanced loss function with Bayes LUE under the quadratic loss function.     

Performance of Preliminary Test Estimator Under Linex Loss Function

This article describes two bivariate geometric distributions. We investigate characterizations of bivariate geometric distributions using conditional failure rates and study properties of the bivariate geometric distributions. The bivariate models are fitted to real-life data using the Method of Moments, Maximum Likelihood, and Bayes Estimators. Two methods of moments estimators, in each bivariate geometric model, are compared and evaluated for their performance in terms of bias vector and covariance matrix. This comparison is done through a Monte Carlo simulation. Chi-square goodness-of-fit tests are used to evaluate model performance.     

Estimation of the Scale Parameter of the Selected Gamma Population Under the Entropy Loss Function

Let X ₁, X ₂,…, X _k be k (≥2) independent random variables from gamma populations Π₁, Π₂,…, Π _k with common known shape parameter α and unknown scale parameter θ _i , i = 1,2,…,k, respectively. Let X _(i) denotes the ith order statistics of X ₁,X ₂,…,X _k . Suppose the population corresponding to largest X _(k) (or the smallest X ₍₁₎) observation is selected. We consider the problem of estimating the scale parameter θ _M (or θ _J ) of the selected population under the entropy loss function. For k ≥ 2, we obtain the Unique Minimum Risk Unbiased (UMRU) estimator of θ _M (and θ _J ). For k = 2, we derive the class of all linear admissible estimators of the form cX ₍₂₎ (and cX ₍₁₎) and show that the UMRU estimator of θ _M is inadmissible. The results are extended to some subclass of exponential family.     

Bayes Estimation of Intraclass Correlation Coefficients Under Unequal Family Sizes

Bayesian inference for the intraclass correlation ρ is considered under unequal family sizes. We obtain the posterior distribution of ρ and then compare the performance of the Bayes estimator (posterior mean of ρ) with that of Srivastava's (1984) estimator through simulation. Simulation study shows that the Bayes estimator performs better than the Srivastava's estimator in terms of lower mean square error. We also obtain large sample posteriors of ρ based on the asymptotic posterior distribution and based on the Laplace approximation.     

Bayesian inference for the randomly censored Weibull distribution

In this paper, we consider the Bayesian inference of the unknown parameters of the randomly censored Weibull distribution. A joint conjugate prior on the model parameters does not exist; we assume that the parameters have independent gamma priors. Since closed-form expressions for the Bayes estimators cannot be obtained, we use Lindley's approximation, importance sampling and Gibbs sampling techniques to obtain the approximate Bayes estimates and the corresponding credible intervals. A simulation study is performed to observe the behaviour of the proposed estimators. A real data analysis is presented for illustrative purposes.     

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.     

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

In this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.     

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.     

Shrinkage Testimators for the Shape Parameter of Pareto Distribution Using LINEX Loss Function

In this article, shrinkage testimators for the shape parameter of a Pareto distribution are considered, when its prior guess value is available. The choices of shrinkage factor are also suggested. The proposed testimators are compared with the minimum risk estimator among the class of unbiased estimators with the LINEX loss function.     

Bayes estimation of Gompertz distribution parameters and acceleration factor under partially accelerated life tests with type-I censoring

In this paper, the Bayesian approach is applied to the estimation problem in the case of step stress partially accelerated life tests with two stress levels and type-I censoring. Gompertz distribution is considered as a lifetime model. The posterior means and posterior variances are derived using the squared-error loss function. The Bayes estimates cannot be obtained in explicit forms. Approximate Bayes estimates are computed using the method of Lindley [D.V. Lindley, Approximate Bayesian methods, Trabajos Estadistica 31 (1980), pp. 223–237]. The advantage of this proposed method is shown. The approximate Bayes estimates obtained under the assumption of non-informative priors are compared with their maximum likelihood counterparts using Monte Carlo simulation.