Robust nonparametric estimation of the intensity function of point data

Abstract

Intensity functions—which describe the spatial distribution of the occurrences of point processes—are useful for risk assessment. This paper deals with the robust nonparametric estimation of the intensity function of space–time data from events such as earthquakes. The basic approach consists of smoothing the frequency histograms with the local polynomial regression (LPR) estimator. This method allows for automatic boundary corrections, and its jump-preserving ability can be improved with robustness. We derive a robust local smoother from the weighted-average approach to M-estimation and we select its bandwidths with robust cross-validation (RCV). Further, we develop a robust recursive algorithm for sequential processing of the data binned in time. An extensive application to the Northern California earthquake catalog in the San Francisco, CA, area illustrates the method and proves its validity.