Some first inferential tools for spatial regression with differential regularization

Spatial regression with differential regularization is an innovative approach at the crossroad between functional data analysis and spatial data analysis. These models have been shown to be numerically efficient and capable to handle complex applied problems. On the other hand, their theoretical properties are still largely unexplored. Here we consider the discrete estimators in spatial regression models with differential regularization, obtained after numerical discretization, using an expansion on a finite element basis. We study the consistency and the asymptotic normality of these discrete estimators. We also propose a nonparametric test procedure for the linear part of the models, based on random sign-flipping of the score components. The test exploits an appropriate decomposition of the smoothing matrix, in order to reduce the effect of the spatial dependence, without any parametric assumption on the form of the correlation structure. The proposed test is shown to be superior to parametric alternatives.