Statistical estimation of jump rates for a specific class of Piecewise Deterministic Markov Processes

TL;DR

This paper develops a nonparametric estimator for the jump rate of a specific class of Piecewise Deterministic Markov Processes, demonstrating its asymptotic efficiency and illustrating its performance through simulations.

Contribution

It introduces a novel nonparametric estimation method for jump rates in PDMPs with increasing deterministic motion and shrinking jumps, applicable to models like TCP window size.

Findings

Estimator's squared-loss error rate is close to n^{-s/(2s+1)} for Hölder smoothness s.

Simulation results confirm the estimator's effectiveness.

Applicable to models like TCP window size and bacterial cell size.

Abstract

We consider the class of Piecewise Deterministic Markov Processes (PDMP), whose state space is $\R\_{+}^{*}$, that possess an increasing deterministic motion and that shrink deterministically when they jump. Well known examples for this class of processes are Transmission Control Protocol (TCP) window size process and the processes modeling the size of a "marked" {\it Escherichia coli} cell. Having observed the PDMP until its $n$th jump, we construct a nonparametric estimator of the jump rate $\lambda$. Our main result is that for $D$ a compact subset of $\R\_{+}^{*}$, if $\lambda$ is in the H{\''{o}}lder space ${\mathcal H}^s({\mathcal D})$, the squared-loss error of the estimator is asymptotically close to the rate of $n^{-s/(2s+1)}$. Simulations illustrate the behavior of our estimator.