Block-band behavior of spatial correlations: An analytical asymptotic study in a spatial exponential family data setup

There is a long history of spatial regression analysis where it is important to accommodate the spatial correlations among the responses from neighboring locations for any valid inferences. Among numerous modeling approaches, the so-called spatial auto-regression (SAR) model in a linear setup, and the conditional auto-regression (CAR) model in a binary setup, are widely used. For spatial binary analysis, there exists two other competitive approaches, namely the bivariate probit models (BPM) based composite likelihood approach using local lattices; and a ‘Working’ correlations based QL (quasi-likelihood) (WCQL) approach. These correlation models, however, fail to accommodate both within and between correlations among spatial families, where a spatial family is naturally formed within a threshold distance of a selected location, and the member locations between two neighboring families may also be correlated. In this paper, we exploit this latter two-ways, within and between correlations among spatial families and develop a unified correlation model for all exponential family based such as linear, count or binary data. We further exploit the proposed correlation structure based generalized quasi-likelihood (GQL) and method of moments (MM) approaches for model parameters estimation. As far as the estimation properties are concerned, because in practice one encounters a large spatial sample, we make sure that the proposed GQL and MM estimators are consistent.