Local based approach is a major category of methods for spatial outlier detection (SOD). Currently, there is a lack of systematic analysis on the statistical properties of this framework. For example, most methods assume identical and independent normal distributions (i.i.d. normal) for the calculated local differences, but no justifications for this critical assumption have been presented. The methods' detection performance on geostatistic data with linear or nonlinear trend is also not well studied. In addition, there is a lack of theoretical connections and empirical comparisons between local and global based SOD approaches. This paper discusses all these fundamental issues under the proposed Generalized Local Statistical (GLS) framework. Furthermore, robust estimation and outlier detection methods are designed for the new GLS model. Extensive simulations demonstrated that the SOD method based on the GLS model significantly outperformed all existing approaches when the spatial data exhibits a linear or nonlinear trend.