Generalized Maximum Likelihood Method in Linear Mixed Models with an Application in Small-Area Estimation

Standard methods frequently produce zero estimates of dispersion parameters in the underlying linear mixed model. As a consequence, the EBLUP estimate of a small area mean reduces to a simple regression estimate. In this paper, we consider a class of generalized maximum residual likelihood estimators that covers the well-known profile maximum likelihood and the residual maximum likelihood estimators. The general class of estimators has a number of different estimators for the dispersion parameters that are strictly positive and enjoy good asymptotic properties. In addition, the mean squared error of the corresponding EBLUP estimator is asymptotically equivalent to those of the profile maximum likelihood and residual maximum likelihood estimators in the higher order asymptotic sense. However, the strictly positive generalized maximum likelihood estimators have an advantage over the standard methods in estimating the shrinkage parameters and in constructing the parametric bootstrap prediction intervals of the small area means. We shall illustrate our methodology using an example from the SAIPE program of the US Census Bureau.