A nonparametric test for Cox processes

TL;DR

This paper introduces two nonparametric tests to determine if a Cox process is Poisson, based on empirical mean and variance comparisons, with proven asymptotic properties and demonstrated effectiveness on real data.

Contribution

It extends classical overdispersion tests to a functional setting for Cox processes, providing a practical, covariate-free testing procedure with theoretical and empirical validation.

Findings

Tests have good empirical performance.

Asymptotic distributions are derived.

Applications to real data demonstrate usefulness.

Abstract

In a functional setting, we propose two test statistics to highlight the Poisson nature of a Cox process when n copies of the process are available. Our approach involves a comparison of the empirical mean and the empirical variance of the functional data and can be seen as an extended version of a classical overdispersion test for counting data. The limiting distributions of our statistics are derived using a functional central limit theorem for c`adl`ag martingales. We also study the asymptotic power of our tests under some local alternatives. Our procedure is easily implementable and does not require any knowledge of covariates. A numerical study reveals the good performances of the method. We also present two applications of our tests to real data sets.