Search

For the hierarchical uniform and inverse gamma model, we calculate the Bayes posterior estimator of the parameter of the uniform 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 function and the corresponding PESL. Moreover, we obtain empirical Bayes estimators of the parameter of the uniform distribution by two methods. Note that the estimators of the hyperparameters of the model by the Maximum Likelihood Estimation (MLE) method are summarized in a theorem, whose proof involves the upper incomplete gamma function and a special case of the Meijer G-function. In numerical simulations, we address from four perspectives. First, we exemplify the two inequalities of the Bayes posterior estimators and the PESLs. Second, we illustrate that the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters. Third, we calculate the goodness-of-fit of the model for the simulated data. Fourth, we plot the marginal densities of the model for various hyperparameters. Finally, we utilize the current prices of the 300 component stocks of Shenzhen 300 Index to illustrate our theoretical studies. [ABSTRACT FROM AUTHOR]

For the hierarchical normal and inverse gamma model, we calculate the Bayes posterior estimator of the variance parameter of the normal 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 variance parameter under the squared error loss function and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the variance parameter of the normal distribution with a conjugate inverse gamma prior by two methods. In numerical simulations, we have illustrated five aspects: 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 comparisons of the Bayes posterior estimators and the PESLs of the oracle, moment, and MLE methods; and the plots of the marginal densities for various hyperparameters. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we utilize the %bodyfat data of 250 men of various ages to illustrate our theoretical studies. [ABSTRACT FROM AUTHOR]

For the hierarchical inverse gamma and inverse gamma model, we calculate the Bayes posterior estimator of the rate parameter of the inverse gamma 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 rate parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the rate parameter of the inverse gamma distribution with a conjugate inverse gamma prior by the moment and MLE methods. In numerical simulations, we have illustrated four aspects: Consistency, goodness-of-fit, comparison, and marginal densities. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, the model could potentially be used to fit right skewed data, not left-skewed data. [ABSTRACT FROM AUTHOR]

Two-stage hierarchical models have been widely used in small area estimation to produce indirect estimates of areal means. When the areas are treated exchangeably and the model parameters are assumed to be the same over all areas, we might lose the efficiency in the presence of spatial heterogeneity. To overcome this problem, we consider a two-stage area-level model based on natural exponential family with spatially varying model parameters. We employ geographically weighted regression approach to estimating the varying parameters and suggest a new empirical Bayes estimator of the areal mean. We also discuss some related problems, including the mean squared error estimation, benchmarked estimation and estimation in non-sampled areas. The performance of the proposed method is evaluated through simulations and applications to two data sets. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Most of the samples in the real world are from the normal distributions with unknown mean and variance, for which it is common to assume a conjugate normal-inverse-gamma prior. We calculate the empirical Bayes estimators of the mean and variance parameters of the normal distribution with a conjugate normal-inverse-gamma prior by the moment method and the Maximum Likelihood Estimation (MLE) method in two theorems. After that, we illustrate the two theorems for the monthly simple returns of the Shanghai Stock Exchange Composite Index. [ABSTRACT FROM AUTHOR]

This article considers an empirical Bayes testing problem for the guarantee lifetime in the two-parameter exponential distributions with non identical components. We study a method of constructing empirical Bayes tests under a class of unknown prior distributions for the sequence of the component testing problems. The asymptotic optimality of the sequence of empirical Bayes tests is studied. Under certain regularity conditions on the prior distributions, it is shown that the sequence of the constructed empirical Bayes tests is asymptotically optimal, and the associated sequence of regrets converges to zero at a rateO(n− 1 + 1/[2(r + α) + 1]) for some integerr⩾ 0 and 0 ⩽ α ⩽ 1 depending on the unknown prior distributions, wherenis the number of past data available when the (n+ 1)st component testing problem is considered. [ABSTRACT FROM PUBLISHER]

The problem of selecting a population according to "selection and ranking" is an important statistical problem. The ideas in selecting the best populations with some demands having optimal criterion have been suggested originally by Bechhofer (1954) and Gupta (1956, 1965). In the area of ranking and selection, the large part of literature is connected with a single criterion. However, this may not satisfy the experimenter's demand. We followmethodology of Huang and Lai (1999) and themain focus of this article is to select a best population under Type-II progressively censored data for the case of right tail exponential distributions with a bounded and unbounded supports forμi. We formulate the problem and develop a Bayesian setup with two kinds of bounded and unbounded prior forμi. We introduce an empirical Bayes procedure and study the large sample behavior of the proposed rule. It is shown that the proposed empirical Bayes selection rule is asymptotically optimal. [ABSTRACT FROM AUTHOR]

We study empirical and hierarchical Bayes approaches to the problem of estimating an infinite-dimensional parameter in mildly ill-posed inverse problems. We consider a class of prior distributions indexed by a hyperparameter that quantifies regularity. We prove that both methods we consider succeed in automatically selecting this parameter optimally, resulting in optimal convergence rates for truths with Sobolev or analytic 'smoothness', without using knowledge about this regularity. Both methods are illustrated by simulation examples. [ABSTRACT FROM AUTHOR]

This article deals with the empirical Bayes estimation of the parameter θ in a uniform distribution U(0, θ) based on randomly right censored data. By mimicking the form of the Bayes estimator, an empirical Bayes estimator [image omitted] is constructed. The asymptotic optimality of [image omitted] is investigated. It is shown that under certain conditions, [image omitted] is asymptotically optimal with a rate of convergence n-λr/2(r+1), where n is the number of past data available when the present estimation problem is considered, and 0 < λ ≤2, and r is a positive integer related to some conditions. [ABSTRACT FROM AUTHOR]

We study empirical Bayes estimation of reliability in a lifetime distribution having probability density f(y|θ) = αyαν-1 exp(-yα/θ)/[Γ(ν)θν], where 0 < θ1 < θ < θ2 < ∞ for some known constants θ1 and θ2. An empirical Bayes estimator [image omitted] is constructed and its associated asymptotic optimality is studied. It is shown that [image omitted] is asymptotically optimal, and the regret of [image omitted] converges to zero at a rates O(ln2n/n) (for 0 < ν < 1/2 or ν = 1 cases) or O(ln2ν+ν* n/n) (for 1/2 ≤ ν < 1 or ν > 1 and ν* = max(1, ν) cases), where n is the number of past data available when the estimation problem is considered. [ABSTRACT FROM AUTHOR]

Group testing has been long recognized as an efficient method to classify all the experimental units into two mutually exclusive categories: defective or not defective. In recent years, more attention has been brought to the estimation of the population prevalence rate p of a disease, or of some property, using group testing. In this article, we propose two scaled squared-error loss functions, which improve the Bayesian approach to estimating p in terms of minimizing the mean squared error (MSE) of the Bayes estimators of p for small p. We show that the new estimators are preferred over the estimator from the usual squared-error loss function and the maximum likelihood estimator (MLE) for small p. [ABSTRACT FROM AUTHOR]

Record values can be viewed as order statistics from a sample whose size is determined by the values and the order of occurrence of observations. Bayes and empirical Bayes estimators for the unknown parameter of the generalized exponential distribution are derived based on record statistics. These estimates are obtained based on squared error and LINEX loss functions. Prediction bounds for future lower record values are obtained by using Bayes and empirical Bayes techniques. Numerical example is given to illustrate the results. [ABSTRACT FROM AUTHOR]