A New Adjusted Residual Likelihood Method for the Fay-Herriot Small Area Model

In the context of the Fay-Herriot model, a mixed regression model routinely used to combine information from various sources in small area estimation, certain adjustments to a standard likelihood (e.g., profile, residual, etc.) have been recently proposed in order to produce strictly positive and consistent model variance estimators. These adjustments protect the resulting empirical best linear unbiased prediction (EBLUP) estimator of a small area mean from possible over-shrinking to the regression estimator. However, the existing adjusted likelihood methods can lead to high bias in the estimation of both model variance and the associated shrinkage factors and can produce a negative second-order unbiased mean square error (MSE) estimate of an EBLUP. In this paper, we propose a new adjustment factor that rectifies the above-mentioned problems associated with the existing adjusted likelihood methods. In particular, we show that our proposed adjusted residual maximum likelihood estimators of the model variance and the shrinkage factors enjoy the same higher-order asymptotic bias properties of the corresponding residual maximum likelihood estimators. We compare performances of the proposed method with the existing methods using Monte Carlo simulations.