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 the possible over-shrinking to the regression estimator. However, in certain cases, the existing adjusted likelihood methods can lead to high biases in the estimation of both model variance and the associated shrinkage factors and can even 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 and profile 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 and profile maximum likelihood estimators, respectively. We compare performances of the proposed method with the existing methods using Monte Carlo simulations.
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