scholarly journals On empirical Bayes estimation of multivariate regression coefficient

2006 ◽  
Vol 2006 ◽  
pp. 1-18
Author(s):  
R. J. Karunamuni ◽  
L. Wei
Keyword(s):  
Multivariate Regression ◽  
Empirical Bayes ◽  
Mean Squared Error ◽  
Bayes Estimator ◽  
Bayes Estimation ◽  
Empirical Bayes Estimation ◽  
Squared Error Loss ◽  
Squared Error ◽  
The Mean ◽  
Empirical Bayes Estimator

We investigate the empirical Bayes estimation problem of multivariate regression coefficients under squared error loss function. In particular, we consider the regression modelY=Xβ+ε, whereYis anm-vector of observations,Xis a knownm×kmatrix,βis an unknownk-vector, andεis anm-vector of unobservable random variables. The problem is squared error loss estimation ofβbased on some “previous” dataY1,…,Ynas well as the “current” data vectorYwhenβis distributed according to some unknown distributionG, whereYisatisfiesYi=Xβi+εi,i=1,…,n. We construct a new empirical Bayes estimator ofβwhenεi∼N(0,σ2Im),i=1,…,n. The performance of the proposed empirical Bayes estimator is measured using the mean squared error. The rates of convergence of the mean squared error are obtained.

scholarly journals Robust empirical Bayes small area estimation with density power divergence

Biometrika ◽  
2020 ◽  
Vol 107 (2) ◽  
pp. 467-480 ◽  
Author(s):  
S Sugasawa
Keyword(s):  
Small Area ◽  
Empirical Bayes ◽  
Mean Squared Error ◽  
Bayes Estimator ◽  
Model Parameters ◽  
Density Power Divergence ◽  
Simple Modification ◽  
Squared Error ◽  
Power Divergence ◽  
Empirical Bayes Estimator

Summary A two-stage normal hierarchical model called the Fay–Herriot model and the empirical Bayes estimator are widely used to obtain indirect and model-based estimates of means in small areas. However, the performance of the empirical Bayes estimator can be poor when the assumed normal distribution is misspecified. This article presents a simple modification that makes use of density power divergence and proposes a new robust empirical Bayes small area estimator. The mean squared error and estimated mean squared error of the proposed estimator are derived based on the asymptotic properties of the robust estimator of the model parameters. We investigate the numerical performance of the proposed method through simulations and an application to survey data.

5. Exploiting Spatial Dependence to Improve Measurement of Neighborhood Social Processes

2009 ◽  
Vol 39 (1) ◽  
pp. 151-183 ◽  
Author(s):  
Natalya Verbitsky Savitz ◽  
Stephen W. Raudenbush
Keyword(s):  
Spatial Dependence ◽  
Empirical Bayes ◽  
Mean Squared Error ◽  
Ordinary Least Squares ◽  
Social Processes ◽  
Bayes Estimator ◽  
Random Errors ◽  
Improve Measurement ◽  
Empirical Bayes Estimator ◽  
Bayes Approach

A number of recent studies have used surveys of neighborhood informants and direct observation of city streets to assess aspects of community life such as collective efficacy, the density of kin networks, and social disorder. Raudenbush and Sampson (1999a) have coined the term “ecometrics” to denote the study of the reliability and validity of such assessments. Random errors of measurement will attenuate the associations between these assessments and key outcomes. To address this problem, some studies have used empirical Bayes methods to reduce such biases, while assuming that neighborhood random effects are statistically independent. In this paper we show that the precision and validity of ecometric measures can be considerably improved by exploiting the spatial dependence of neighborhood social processes within the framework of empirical Bayes shrinkage. We compare three estimators of a neighborhood social process: the ordinary least squares estimator (OLS), an empirical Bayes estimator based on the independence assumption (EBE), and an empirical Bayes estimator that exploits spatial dependence (EBS). Under our model assumptions, EBS performs better than EBE and OLS in terms of expected mean squared error loss. The benefits of EBS relative to EBE and OLS depend on the magnitude of spatial dependence, the degree of neighborhood heterogeneity, as well as neighborhood's sample size. A cross-validation study using the original 1995 data from the Project on Human Development in Chicago Neighborhoods and a replication of that survey in 2002 show that the empirical benefits of EBS approximate those expected under our model assumptions; EBS is more internally consistent and temporally stable and demonstrates higher concurrent and predictive validity. A fully Bayes approach has the same properties as does the empirical Bayes approach, but it is preferable when the number of neighborhoods is small.

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