Adaptive tests of homogeneity for a Poisson process

TL;DR

This paper introduces adaptive, nonparametric tests for homogeneity in Poisson processes, achieving near-optimal separation rates across various Besov function classes with theoretical guarantees and Monte Carlo validation.

Contribution

It develops new adaptive testing procedures for Poisson process homogeneity that are optimal over multiple Besov classes, combining model selection and thresholding techniques.

Findings

Tests achieve near-minimax separation rates.

Procedures are adaptive over multiple Besov classes.

Monte Carlo simulations confirm test power.

Abstract

We propose to test the homogeneity of a Poisson process observed on a finite interval. In this framework, we first provide lower bounds for the uniform separation rates in $\mathbb{L}^2$ norm over classical Besov bodies and weak Besov bodies. Surprisingly, the obtained lower bounds over weak Besov bodies coincide with the minimax estimation rates over such classes. Then we construct non asymptotic and nonparametric testing procedures that are adaptive in the sense that they achieve, up to a possible logarithmic factor, the optimal uniform separation rates over various Besov bodies simultaneously. These procedures are based on model selection and thresholding methods. We finally complete our theoretical study with a Monte Carlo evaluation of the power of our tests under various alternatives.