Small Area Estimation

A Brief Primer on Important Terms in Small Area Estimation

For m small areas, suppose Yij,J = 1,Ni denote values of a response variable (Y) for the Ni units in the ith small area. Imagine one would like to estimate the finite population mean. Suppose X is a vector of explanatory variables. If explanatory variables are available for all the sampled units in the ith small area, to be denoted for simplicity by 1, …,ni, then a unit-level model is used. But if only direct estimates Yi for yi and summary data x; for explanatory variables are available at the small area level, then an area-level model is used. If indirect small area estimates are produced by fitting a model relating the response variable and explanatory variables, and prediction of a small area mean is obtained by substituting explanatory variables into the estimated model, one gets a synthetic estimate, denoted by ŷ. Synthetic estimates are much too model dependent, susceptible to model failure, and not design-consistent. A composite estimate, which is a convex combination of yi and ŷis, rectifies these deficiencies. This entry considers only some of the basic aspects of small area estimation. For example, neither the time series and cross-sectional approach to small area estimation nor the interval estimation problem is considered here. For this and many other important topics, the advanced reader should consult J. N. K. Rao's Small Area Estimation.

Two Popular Small Area Models

Both linear and nonlinear models and both Bayesian and frequentist approaches are popular in small area estimation. While the estimated best linear unbiased prediction (EBLUP) approach is key to developing composite estimates based on mixed linear models, the empirical Bayes (EB) and the hierarchical Bayes (HB) approaches can be used for both linear and nonlinear models. Many model-based developments in small area estimation use normality assumptions. For normal linear models, the EBLUP and EB predictors of the small area means are identical.

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