Constrained Bayes and Empirical Bayes Estimation with Balanced Loss Functions

This article develops constrained Bayes and empirical Bayes estimators under balanced loss functions. In the normal-normal example, estimators of the mean squared errors of the EB and constrained EB estimators are provided which are correct asymptotically up to O(m ^?1), m denoting the number of strata.     

SMALL AREA ESTIMATION USING SURVEY WEIGHTS WITH FUNCTIONAL MEASUREMENT ERROR IN THE COVARIATE

Nested error linear regression models using survey weights have been studied in small area estimation to obtain efficient model‐based and design‐consistent estimators of small area means. The covariates in these nested error linear regression models are not subject to measurement errors. In practical applications, however, there are many situations in which the covariates are subject to measurement errors. In this paper, we develop a nested error linear regression model with an area‐level covariate subject to functional measurement error. In particular, we propose a pseudo‐empirical Bayes (PEB) predictor to estimate small area means. This predictor borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. We also employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator.     

Prediction Error of Small Area Predictors Shrinking Both Means and Variances

The article considers a new approach for small area estimation based on a joint modelling of mean and variances. Model parameters are estimated via expectation–maximization algorithm. The conditional mean squared error is used to evaluate the prediction error. Analytical expressions are obtained for the conditional mean squared error and its estimator. Our approximations are second‐order correct, an unwritten standardization in the small area literature. Simulation studies indicate that the proposed method outperforms the existing methods in terms of prediction errors and their estimated values.     

The Superiorities of Bayes Linear Minimum Risk Estimation in Linear Model

In this article, the Bayes linear minimum risk estimator (BLMRE) of parameters is derived in linear model. The superiorities of the BLMRE over ordinary least square estimator (LSE) is studied in terms of the mean square error matrix (MSEM) criterion and Pitman closeness (PC) criterion.     

A BLUP Synthetic Versus an EBLUP Estimator: An Empirical Study of a Small Area Estimation Problem

Model-based estimators are becoming very popular in statistical offices because Governments require accurate estimates for small domains that were not planned when the study was designed, as their inclusion would have produced an increase in the cost of the study. The sample sizes in these domains are very small or even zero; consequently, traditional direct design-based estimators lead to unacceptably large standard errors. In this regard, model-based estimators that 'borrow information' from related areas by using auxiliary information are appropriate. This paper reviews, under the model-based approach, a BLUP synthetic and an EBLUP estimator. The goal is to obtain estimators of domain totals when there are several domains with very small sample sizes or without sampled units. We also provide detailed expressions of the mean squared error at different levels of aggregation. The results are illustrated with real data from the Basque Country Business Survey.     

Pseudo-empirical Bayes estimation of small area means under a nested error linear regression model with functional measurement errors

Small area estimation is studied under a nested error linear regression model with area level covariate subject to measurement error. Ghosh and Sinha (2007) obtained a pseudo-Bayes (PB) predictor of a small area mean and a corresponding pseudo-empirical Bayes (PEB) predictor, using the sample means of the observed covariate values to estimate the true covariate values. In this paper, we first derive an efficient PB predictor by using all the available data to estimate true covariate values. We then obtain a corresponding PEB predictor and show that it is asymptotically “optimal”. In addition, we employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator. Our results show that the proposed PEB predictor can lead to significant gain in efficiency over the previously proposed PEB predictor. Area level models are also studied.     

Empirical Bayes Confidence Intervals for Means of Natural Exponential Family-Quadratic Variance Function Distributions with Application to Small Area Estimation

Abstract.  The paper develops empirical Bayes (EB) confidence intervals for population means with distributions belonging to the natural exponential family-quadratic variance function (NEF-QVF) family when the sample size for a particular population is moderate or large. The basis for such development is to find an interval centred around the posterior mean which meets the target coverage probability asymptotically, and then show that the difference between the coverage probabilities of the Bayes and EB intervals is negligible up to a certain order. The approach taken is Edgeworth expansion so that the sample sizes from the different populations need not be significantly large. The proposed intervals meet the target coverage probabilities asymptotically, and are easy to construct. We illustrate use of these intervals in the context of small area estimation both through real and simulated data. The proposed intervals are different from the bootstrap intervals. The latter can be applied quite generally, but the order of accuracy of these intervals in meeting the desired coverage probability is unknown.     

Estimation of Standard Errors of Empirical Bayes Estimators in Capm-Type Models

This paper is concerned with the problem of estimating the standard errors of the empirical Bayes estimators in linear regression models. The problem of deriving an exact expression for the standard error of this estimator is generally intractable. We suggest a procedure based on Efron’s bootstrap method as a way of estimating the standard error. It is shown, through simulations, that the bootstrap method provides a more accurate estimate of the standard error of the empirical Bayes estimator than the traditional large sample method.     

On Mean Squared Prediction Error Estimation in Small Area Estimation Problems

For a general linear mixed normal model, a new linearized weighted jackknife method is proposed to estimate the mean squared prediction error (MSPE) of an empirical best linear unbiased predictor (EBLUP) of a general mixed effect. Different MSPE estimators are compared using a Monte Carlo simulation study.     

A Small Area Predictor under Area-Level Linear Mixed Models with Restrictions

A calibrated small area predictor based on an area-level linear mixed model with restrictions is proposed. It is showed that such restricted predictor, which guarantees the concordance between the small area estimates and a known estimate at the aggregate level, is the best linear unbiased predictor. The mean squared prediction error of the calibrated predictor is discussed. Further, a restricted predictor under a particular time-series and cross-sectional model is presented. Within a simulation study based on real data collected from a longitudinal survey conducted by a national statistical office, the proposed estimator is compared with other competitive restricted and non-restricted predictors.     

Empirical and Hierarchical Bayesian Estimation in Finite Population Sampling under Structural Measurement Error Models

Abstract.  This paper considers simultaneous estimation of means from several strata. A model-based approach is taken, where the covariates in the superpopulation model are subject to measurement errors. Empirical Bayes (EB) and Hierarchical Bayes estimators of the strata means are developed and asymptotic optimality of EB estimators is proved. Their performances are examined and compared with that of the sample mean in a simulation study as well as in data analysis.     

Quasi-Minimax Estimation in the Linear Model with Measurement Errors

In this article, we consider quasi-minimax estimation in the linear regression model where some covariates are measured with additive errors. When measurement errors are directly ignored the minimax risk of the resulting estimator can be large. By correcting the attenuation we propose a penalized quadratic risk function. A simulation study is conducted to illustrate the performance of the proposed estimators.     

Objective Priors for Parameters in a Normal Linear Regression with Measurement Error

We investigate certain objective priors for the parameters in a normal linear regression models with one of the explanatory variables subject to measurement error. We first show that the use of the standard non informative prior for normal linear regression without measurement error leads to an improper posterior in the measurement error model. We then derive the Jeffreys prior and reference priors, and show that they lead to proper posteriors. We use simulation study to compare the frequentist performance of the estimates derived using these priors, and the MLE.     

Estimation of Slope for Linear Regression Model with Uncertain Prior Information and Student-t Error

This article considers estimation of the slope parameter of the linear regression model with Student-t errors in the presence of uncertain prior information on the value of the unknown slope. Incorporating uncertain non sample prior information with the sample data the unrestricted, restricted, preliminary test, and shrinkage estimators are defined. The performances of the estimators are compared based on the criteria of unbiasedness and mean squared errors. Both analytical and graphical methods are explored. Although none of the estimators is uniformly superior to the others, if the non sample information is close to its true value, the shrinkage estimator over performs the rest of the estimators.     

Bayes and Empirical Bayes Estimation of the Parameter n in a Binomial Distribution

The problem of estimating the total number of trials n in a binomial distribution is reconsidered in this article for both cases of known and unknown probability of success p from the Bayesian viewpoint. Bayes and empirical Bayes point estimates for n are proposed under the assumption of a left-truncated prior distribution for n and a beta prior distribution for p. Simulation studies are provided in this article in order to compare the proposed estimate with the most familiar n estimates.     

Finite-Sample Properties of the Maximum Likelihood Estimator for the Poisson Regression Model With Random Covariates

We examine the small-sample behaviour of the maximum likelihood estimator for the Poisson regression model with random covariates. Analytic expressions for the second-order bias and mean squared error are derived, and we undertake some numerical evaluations to illustrate these results for the single covariate case. The properties of the bias-adjusted maximum likelihood estimator are investigated in a Monte Carlo experiment. Correcting the estimator for its second-order bias is found to be effective in the cases considered, and we recommend its use when the Poisson regression model is estimated by maximum likelihood with small samples.     

Model Reduction for Prediction in Regression Models

We look at prediction in regression models under squared loss for the random x case with many explanatory variables. Model reduction is done by conditioning upon only a small number of linear combinations of the original variables. The corresponding reduced model will then essentially be the population model for the chemometricians' partial least squares algorithm. Estimation of the selection matrix under this model is briefly discussed, and analoguous results for the case with multivariate response are formulated. Finally, it is shown that an assumption of multinormality may be weakened to assuming elliptically symmetric distribution, and that some of the results are valid without any distributional assumption at all.     

Empirical Likelihood Ratio Test for a Change-Point in Linear Regression Model

A nonparametric method based on the empirical likelihood is proposed to detect the change-point in the coefficient of linear regression models. The empirical likelihood ratio test statistic is proved to have the same asymptotic null distribution as that with classical parametric likelihood. Under some mild conditions, the maximum empirical likelihood change-point estimator is also shown to be consistent. The simulation results show the sensitivity and robustness of the proposed approach. The method is applied to some real datasets to illustrate the effectiveness.     

Akaike Information Criterion for Selecting Variables in the Nested Error Regression Model

The Akaike Information Criterion (AIC) is developed for selecting the variables of the nested error regression model where an unobservable random effect is present. Using the idea of decomposing the likelihood into two parts of “within” and “between” analysis of variance, we derive the AIC when the number of groups is large and the ratio of the variances of the random effects and the random errors is an unknown parameter. The proposed AIC is compared, using simulation, with Mallows' C _p , Akaike's AIC, and Sugiura's exact AIC. Based on the rates of selecting the true model, it is shown that the proposed AIC performs better.     

M-Estimator of a Generalized Linear Model with Measurement Errors

In this article, we propose a generalized linear model and estimate the unknown parameters using robust M-estimator. Under suitable conditions and by the strong law of large numbers and central limits theorem, the proposed M-estimators are proved to be consistent and asymptotically normal. We also evaluate the finite sample performance of our estimator through a Monte Carlo study.