Liu-type pretest and shrinkage estimation for the conditional autoregressive model.

Spatial regression models have recently received a lot of attention in a variety of fields to address the spatial autocorrelation effect. One important class of spatial models is the Conditional Autoregressive (CA). Theses models have been widely used to analyze spatial data in various areas, as geography, epidemiology, disease surveillance, civilian planning, mapping of poorness signals and others. In this article, we propose the Liu-type pretest, shrinkage and positive shrinkages estimators for the large-scale effect parameter vector of the CA regression model. The set of the proposed estimators are evaluated analytically via their asymptotic bias, quadratic bias, the asymptotic quadratic risks, and numerically via their relative mean squared errors. Our results demonstrate that the proposed estimators are more efficient than Liu-type estimator. To conclude this paper, we apply the proposed estimators to the Boston housing prices data, and applied a bootstrapping technique to evaluate the estimators based on their mean squared prediction error.