Asymptotic Analysis and Practical Evaluation of Jump Rate Estimators in Piecewise-Deterministic Markov Processes

TL;DR

This paper develops a unified theoretical framework for jump rate estimation in PDMPs, providing consistency, asymptotic normality results, and practical comparisons through simulations and real data analysis.

Contribution

It introduces a unified framework for non-parametric jump rate estimation in PDMPs, enabling rigorous comparison of methods and revealing no single method is universally optimal.

Findings

No method outperforms others uniformly within the same model.

Theoretical results on consistency and asymptotic normality are established.

Numerical simulations and real data analysis validate the theoretical insights.

Abstract

Piecewise-deterministic Markov processes (PDMPs) offer a powerful stochastic modeling framework that combines deterministic trajectories with random perturbations at random times. Estimating their local characteristics (particularly the jump rate) is an important yet challenging task. In recent years, non-parametric methods for jump rate inference have been developed, but these approaches often rely on distinct theoretical frameworks, complicating direct comparisons. In this paper, we propose a unified framework to standardize and consolidate state-of-the-art approaches. We establish new results on consistency and asymptotic normality within this framework, enabling rigorous theoretical comparisons of convergence rates and asymptotic variances. Notably, we demonstrate that no single method uniformly outperforms the others, even within the same model. These theoretical insights are validated through numerical simulations using a representative PDMP application: the TCP model. Furthermore, we extend the comparison to real-world data, focusing on cell growth and division dynamics in Escherichia coli. This work enhances the theoretical understanding of PDMP inference while offering practical insights into the relative strengths and limitations of existing methods.