Abstract
In survey sampling, interest often lies in unplanned domains (or small areas), whose sample sizes may be too small to allow for accurate design-based inference. To improve the direct estimates by borrowing strength from similar domains, most small area methods rely on mixed effects regression models. This contribution extends the well known Fay–Herriot model (Fay and Herriot, 1979) within a Bayesian approach in two directions. First, the default normality assumption for the random effects is replaced by a nonparametric specification using a Dirichlet process. Second, uncertainty on variances is explicitly introduced, recognizing the fact that they are actually estimated from survey data. The proposed approach shrinks variances as well as means, and accounts for all sources of uncertainty. Adopting a flexible model for the random effects allows to accommodate outliers and vary the borrowing of strength by identifying local neighbourhoods where the exchangeability assumption holds. Through application to real and simulated data, we investigate the performance of the proposed model in predicting the domain means under different distributional assumptions. We also focus on the construction of credible intervals for the area means, a topic that has received less attention in the literature. Frequentist properties such as mean squared prediction error (MSPE), coverage and interval length are investigated. The experiments performed seem to indicate that inferences under the proposed model are characterised by smaller mean squared error than competing approaches; frequentist coverage of the credible intervals is close to nominal.
Highlights
Introduction and MotivationThe Fay–Herriot ModelThe information content of a sample survey is clearly not limited to the planned domains, and researchers are often interested in obtaining estimates for a whole variety of subpopulations, that we refer to as small areas
Diao et al (2014) focus on deriving accurate CIs for small area means using the Empirical Best Linear Unbiased Predictor (EBLUP) and estimators of mean square prediction error of EBLUPs based on various methods of estimation of model parameters, under the standard Fay–Herriot model
As mentioned in the Introduction, this contribution focuses on two major assumptions underlying the Fay–Herriot model, namely the normality of random effects and the assumption of known sampling variances
Summary
The information content of a sample survey is clearly not limited to the planned domains, and researchers are often interested in obtaining estimates for a whole variety of subpopulations, that we refer to as small areas. As to the second point, whereas for area-level models the distributional assumption on the sampling errors i is usually justified by the properties of the direct estimators θi, the normality assumption for the random effects νi has no justification other than computational convenience This assumption is difficult to detect in practice, since it involves unobservable quantities. Diao et al (2014) focus on deriving accurate CIs for small area means using the EBLUPs and estimators of mean square prediction error of EBLUPs based on various methods of estimation of model parameters, under the standard Fay–Herriot model They investigate robustness to misspecifications of the random component distribution via simulation. Malec and Muller (2008) discuss use of DP in the context of small area estimation Their model is formulated as a unit-level setup in which the county-specific random effects are described by a mixture of Dirichlet processes.
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