Small area estimation: an empirical best linear unbiased prediction approach

In a large scale survey, we are usually concerned with estimation ofsome characteristics of interest for a large area (e.g., a country).But we are frequently interested in estimating similarcharacteristics for a subpopulation using the same survey data. Thedirect survey estimator which utilizes data only from the small areaof interest has been found to be highly unreliable due to smallsample size. Model-based methods have been used in small areaestimation in order to combine information available from the surveydata and various administrative and census data.We study the empirical best linear unbiased prediction (EBLUP) andits inferences under the general Fay-Herriot small area model.Considering that the currently used variance estimation methods could produce zero estimates, we propose the adjusted densitymethod (ADM) following Morris' comments. This new method alwaysproduces positive estimates. Morris only suggested such adjustmentto the restricted maximum likelihood. Asymptotic theory of ADM isunknown. We prove the consistency for the ADM estimator. We alsopropose an alternate consistent ADM estimator by adjusting themaximum likelihood. By comparing these two ADM estimators both intheory and simulation, we find that the ADM estimator using maximumlikelihood is better than the one using the restricted likelihood interms of bias. We provide a concrete proof for the positiveness andconsistency of both ADM estimators.We also propose EBLUP estimator of $\theta_i$ where we use two ADMestimators of $A$. The associated second-order unbiased Taylorlinearization MSE estimators are also proposed.In addition, a new parametric bootstrap prediction interval methodusing ADM estimator is proposed. The positiveness of ADM estimatorsis emphasized in the construction of the prediction interval. Wealso show that the coverage probability of this new method isaccurate up to $O(m^{-3/2})$.Extensive Monte Carlo simulations are conducted. A data analysis forthe SAIPE data set is also presented. The positiveness of ADMestimators plays a vital role here since for this data set themethod-of-moments, REML, ML and FH methods could be all zero. Weobserve that ADM methods produce EBLUP's which generally put moreweights to the direct survey estimates than the correspondingEBLUP's that use the other methods of variance component estimation.