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# A Unified Bayesian Approach to Small Area Estimation of Mean Parameters in Generalized Linear Models

## Patrick J. Farrell, Brenda MacGibbon, and Thomas J. Tomberlin

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In this study, we present a unified Bayesian approach to small area estimation of mean parameters in generalized linear models. The basic idea consists of incorporating into such a model nested random effects that reflect the complex structure of the data in a multistage sample design. However, as compared to the ordinary linear regression model, it is not feasible to obtain a closed form expression for the posterior distribution of the parameters. The approximation most commonly used in empirical Bayes studies is that proposed by Laird (1978), where the posterior is expressed as a multivariate normal distribution having its mean at the mode and covariance matrix equal to the inverse of the information matrix evaluated at the mode. Inspired by the work of Zeger and Karim (1991) and Gu and Li (1998), we also study a stochastic simulation method to approximate the posterior distribution. Alternatively, a hierarchical Bayes approach based on Gibbs sampling similar to Farrell (2000) can also be employed. We present here the results of a Monte Carlo simulation study to compare point and interval estimates of small area proportions based on these three estimation methods.

, 17 pages

### Document

G-2002-53.pdf (90 KB)