An adjusted maximum likelihood method for solving small area estimation problems

Abstract

For the well-known Fay-Herriot small area model, standard variance component estimation methods frequently produce zero estimates of the strictly positive model variance. As a consequence, an empirical best linear unbiased predictor of a small area mean, commonly used in the small area estimation, could reduce to a simple regression estimator, which typically has an overshrinking problem. We propose an adjusted maximum likelihood estimator of the model variance that maximizes an adjusted likelihood defined as a product of the model variance and a standard likelihood (e.g., profile or residual likelihood) function. The adjustment factor was suggested earlier by Carl Morris in the context of approximating a hierarchical Bayes solution where the hyperparameters, including the model variance, are assumed to follow a prior distribution. Interestingly, the proposed adjustment does not affect the mean squared error property of the model variance estimator or the corresponding empirical best linear unbiased predictors of the small area means in a higher order asymptotic sense. However, as demonstrated in our simulation study, the proposed adjustment has a considerable advantage in the small sample inference, especially in estimating the shrinkage parameters and in constructing the parametric bootstrap prediction intervals of the small area means, which require the use of a strictly positive consistent model variance estimate.

A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models

“…This problem has been pointed out by Li and Lahiri (2010) and Morris and Tang (2011) in small area estimation.…”

“…Adjustment for density maximization (ADM; Morris, 2006;Li & Lahiri, 2010;Morris & Tang, 2011) has been proposed for obtaining strictly positive group-level variance estimates in the context of small area estimation. The Fay-Herriot model (1979) considered in these papers is equivalent to random-effects meta-regression, the model in (1) but with covariates.…”

“…Our penalized likelihood approach to avoid boundary estimates for variance parameters in multilevel models turns out to be similar to, but more general than, the independently developed adjustment for density maximization approach by Morris and Tang (2011). They apply the idea to the Fay and Herriot (1979) model for small-area estimation using area-level (group-level) data and focus on the problem of predicting area-level means (see also Li & Lahiri, 2010). In contrast, we consider unit-level data and focus on estimation of the model parameters and standard errors of regression coefficients.…”

Transforming response values in small area prediction

“…(2.5) 22 where z 1 , z 2 ∼ N(0, A), c 1i = √ (1 + a i )/2, and c 2i = √ (1 − a i )/2 for a i = A/(A +D i ). Hence, g 1i (φ) can be easily calculated 23 by generating a large number of random samples of z 1 and z 2 .…”

“…For λ, we treated the four cases λ = 0.1, 0.4, 0.7 22 and 1.0. The covariates x i were initially generated from the uniform distribution on (0, 4) and fixed in simulation runs.…”