Robust Estimation in the Presence of Deviations from Linearity in Small Domain Models

Julie B. Gershunskaya and Terrance D. Savitsky

Abstract

Small domain estimation models, like the Fay-Herriot, often assume a normally distributed latent process centered on a linear mean function. The linearity assumption may be violated for domains that express idiosyncratic phenomena not captured by the predictors. Under a single component normal distribution prior for the random effects, direct sample estimates for those domains would be viewed as if they were outliers with respect to the model, when in fact they may reflect the underlying true population value. The model interpretation is also confounded by the variances of direct sample estimates because, while typically treated as fixed and known, they are estimates and thus contain noise. In this paper, we construct a joint model for the direct estimates and their variances where we replace the normal distribution for the latent process with a nonparametric mixtures of normal distributions with the goal to improve robustness in estimation quality for these idiosyncratic domains. We devise a model-based screening tool that leverages the posterior predictive distribution under the model to nominate domains where the model may not accurately account for deviations from the linearity assumption. Our screening tool nominates a few domains to allow for a focused investigation to determine whether a deviation from linearity is real. The U.S. Bureau of Labor Statistics' Current Employment Statistics (CES) survey publishes monthly employment estimates for domains defined by industry and geography. Model estimation is performed for smaller domains to improve the reliability of the direct estimator. We compare fit performances for our candidate models under data constructed to be similar to the CES and conduct a simulation study to assess the robustness of our candidate models in the presence of deviations from linearity. We apply our model-based screening method and quantify its ability to improve the quality of published estimates.