In a large scale survey, we are usually concerned with estimation of some characteristics of interest for a large area (e.g., a country). But we are frequently interested in estimating similar characteristics for a subpopulation using the same survey data. The direct survey estimator which utilizes data only from the small area of interest has been found to be highly unreliable due to small sample size. Model-based methods have been used in small area estimation in order to combine information available from the survey data and various administrative and census data.

We study the empirical best linear unbiased prediction (EBLUP) and its 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 density method (ADM) following Morris' comments. This new method always produces positive estimates. Morris only suggested such adjustment to the restricted maximum likelihood. Asymptotic theory of ADM is unknown. We prove the consistency for the ADM estimator. We also propose an alternate consistent ADM estimator by adjusting the maximum likelihood. By comparing these two ADM estimators both in theory and simulation, we find that the ADM estimator using maximum likelihood is better than the one using the restricted likelihood in terms of bias. We provide a concrete proof for the positiveness and consistency of both ADM estimators.

We also propose EBLUP estimator of $\theta_i$ where we use two ADM estimators of $A$. The associated second-order unbiased Taylor linearization MSE estimators are also proposed.

In addition, a new parametric bootstrap prediction interval method using ADM estimator is proposed. The positiveness of ADM estimators is emphasized in the construction of the prediction interval. We also show that the coverage probability of this new method is accurate up to $O(m^{-3/2})$.

Extensive Monte Carlo simulations are conducted. A data analysis for the SAIPE data set is also presented. The positiveness of ADM estimators plays a vital role here since for this data set the method-of-moments, REML, ML and FH methods could be all zero. We observe that ADM methods produce EBLUP's which generally put more weights to the direct survey estimates than the corresponding EBLUP's that use the other methods of variance component estimation.