Cox Point Process Regression

TL;DR

This paper introduces a novel nonparametric regression method for modeling the intensity functions of Cox point processes using covariates and repeated observations, enabling consistent estimation and asymptotic analysis.

Contribution

It proposes a new nonparametric regression approach for Cox point processes that leverages covariates and repeated data to achieve consistent estimation.

Findings

Consistent estimation of intensity functions using covariates and repeated observations.

Derivation of asymptotic convergence rates without parametric assumptions.

Demonstration of the method's applicability to real-world data with covariates.

Abstract

Point processes in time have a wide range of applications that include the claims arrival process in insurance or the analysis of queues in operations research. Due to advances in technology, such samples of point processes are increasingly encountered. A key object of interest is the local intensity function. It has a straightforward interpretation that allows to understand and explore point process data. We consider functional approaches for point processes, where one has a sample of repeated realizations of the point process. This situation is inherently connected with Cox processes, where the intensity functions of the replications are modeled as random functions. Here we study a situation where one records covariates for each replication of the process, such as the daily temperature for bike rentals. For modeling point processes as responses with vector covariates as predictors we propose a novel regression approach for the intensity function that is intrinsically nonparametric. While the intensity function of a point process that is only observed once on a fixed domain cannot be identified, we show how covariates and repeated observations of the process can be utilized to make consistent estimation possible, and we also derive asymptotic rates of convergence without invoking parametric assumptions.