Generalized Spatial Varying-Coefficient Model

Varying-coefficient models are a useful extension of classical linear models. They are very important tools in many scientific areas, such as economics, finance, epidemiology, medical science, ecology and so on. Thanks to their flexibility and interpretability, in the past ten years, the varying-coefficient models have experienced deep and exciting development on both theoretical and applied sides.Up to the present day, most researches about varying-coefficient models are confined to the cases that the coefficients of the models are univariate functions, which are deficient in applications. For example, when we analyze the data sets containing geographical position, the coefficients are bivariate functions. In this paper, We will discuss the multivariate coefficient functions and will get generalized spatial varying-coefficient models.we consider the generalized spatial varying-coefficient models as follows:where Y is a real valued response variable. U = (u₁,…,u_d) and X = (X₁,…, X_p)^T are d-dimensional and p-dimensional random vector respectively.εis a random error withα_j(·) (j = 1,…,p) are unknown functional coefficients with the same smoothness. Locally polynomial fitting has been proved to be a very effective nonparametric method. A major advantage of this method is that it is very simple to visualize how the estimator is using the data when estimating the unknown function at a particular point. First, we use the locally polynomial fitting to estimate the coefficients of the models. Second, in order to evaluate the performance of the estimators, the asymptotical properties (asymptotic bias and asymptotic variance) of the estimators are established. According to the results, under appropriate conditions, the estimators are asymptotically unbiased and converge in mean square to the real value of the coefficients. At the same the time, we proved that the estimators are consistent. Since locally polynomial fitting is a kind of kernel smoothing method, the selection of the bandwidth matrix is crucial, so lastly, we discussed how to choose the bandwidth matrix.