On Consistent Nonparametric Intensity Estimation for Inhomogeneous Spatial Point Processes.

A common nonparametric approach to estimate the intensity function of an inhomogeneous spatial point process is through kernel smoothing. When conducting the smoothing, one typically uses events only in a local set around the point of interest. But the resulting estimator often is inconsistent, because the number of events in a fixed set is of order 1 for spatial point processes. In this article we propose a new covariate-based kernel smoothing method to estimate the intensity function. Our method defines the distance between any two points as the difference between their associated covariate values. Consequently, we determine the kernel weight for a given event of the process as a function of its new distance to the point of interest. Under some suitable conditions on the covariates and the spatial point process, we prove that our new estimator is consistent for the true intensity. To handle the situation with high-dimensional covariates, we also extend sliced inverse regression, a useful dimension-reduction tool in standard regression analysis, to spatial point processes. Simulations and an application to a real data example are used to demonstrate the usefulness of the proposed method. [ABSTRACT FROM AUTHOR]