Estimation of An Optimum Spatial Autocorrelation in Linear Mixed Models Containing Spatial Effects

Fay-Herriot model assumes that the random effects between regions (areas) are independent of each other. This allows the regions to be mutually independent so that the estimators obtained are unbiased estimators. In cases where the regions are not mutually independent, it can develop a model in which the assumptions are violated (not fulfilled) or allow the regions to be dependent. The development of this model is known as small area estimation (SAE) with spatial effects. In small area estimation with spatial effects, one of the important parameters is the spatial autocorrelation parameter. In various small area estimation with spatial effects, it is still very rare to generate estimators of the spatial autocorrelation parameter. Most of the parameter values used are known, that is by trying to enter several spatial autocorrelation parameter values to show that the addition of regional aspects can increase the accuracy of the small area estimation. In addition, they have also tried a restricted maximum likelihood approach in estimating the spatial autocorrelation and component variance but they still assume that the sampling variance is known (assigned). Therefore, this research proposes a concentrated log-likelihood function by means of numerical procedure to find an optimum estimate value for spatial autocorrelation coefficient where both a sampling variance and a component variance are unknown. Parameters estimators obtained in the models, fixed and random effects parameters, are proved to be consistent.