A class of general adjusted maximum likelihood methods for desirable mean squared error estimation of EBLUP under the Fay–Herriot small area model

The empirical best linear unbiased prediction (EBLUP) estimator is utilized for efficient inference in various research areas, especially for small-area estimation. In order to measure its uncertainty, we generally need to estimate its mean squared prediction error (MSE). Ideally, an EBLUP-based method should not only provide a second-order unbiased estimator of MSE of EBLUP but also maintain strict positivity in estimators of both model variance parameter and MSE of EBLUP. Fortunately, the MSE estimators proposed in Yoshimori and Lahiri (2014) and Hirose and Lahiri (2018) achieve the three desired properties simultaneously. As far as we know, no other MSE estimator does so. In this paper, we therefore seek an adequate class of general adjusted maximum-likelihood methods that simultaneously achieve the three desired properties of MSE estimation. To establish that the investigated class does so, we reveal the relationship between the general adjusted maximum-likelihood method for the model variance parameter and the general functional form of the second-order unbiased MSE estimator, maintaining strict positivity. We also compare the performance of several MSE estimators in our investigated class and others through a Monte Carlo simulation study. The results show that the MSE estimators in our investigated class perform better than those in others.