Regular estimation of the intensity function of a point process. Nonparametric methods

The purpose of the present paper is to give an extensive survey of some nonparametric estimation methods for counting process intensities. The methods in the article are based on Aalen’s multiplicative intensity models. After introducing the important tools from the theory of counting processes, the authors briefly comment upon Aalen’s estimator of the integrated intensity function and describe some kernel estimators for the intensity itself obtained by smoothing the increments of the Aalen estimator. Throughout this paper they concentrate on three major approaches for nonparametric estimation: the method of Sieves (A. Karr), the penalized maximum likelihood method (A. Antoniadis and G. Grégoire) and the orthogonal series (least squares) method (I. Mckeague). For each of these cases the authors present the method and summarize the important properties of the derived estimators. The use of these methods for analysing survival data is emphasized. Several other approaches are briefly presented and discussed.