In small area estimation, generalized linear mixed models represent a usefultool for deriving best prediction of counts or proportions. For non-Gaussianresponses, computing the Empirical Best Prediction and the correspondinganalytic approximation to its Mean Squared Error requires the solution of(possibly) multiple integrals that, in general, do not admit closed form. MonteCarlo methods and parametric bootstrap are common choices, even though theircomputational burden represents a non trivial issue. We propose to estimatemodel parameters within a NonParametric Maximum Likelihood framework. In thiscontext, the distribution of the area-specific random parameters is leftunspecified and is approximated by a (discrete) nonparametric distribution.Given the discrete nature of the mixing distribution, we can avoid integralapproximations and considerably reduce the computational effort in the case ofnon-Gaussian responses. Within this framework, we derive the Empirical BestPrediction for a (possibly non-linear) mixed effect and the analyticapproximation of the corresponding MSE. The proposed approach is presented fora general response that belongs to the Exponential Family. Then, we focus onthe relevant case of binary data. The method is tested via a large scalesimulation study and applied to unit-level data from the 2012 Italian LaborForce Survey.
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