Statistical estimation of jump rates for a piecewise deterministic Markov processes with deterministic increasing motion and jump mechanism^∗

Université de Rennes 1, Institut de Recherche mathématique de Rennes, CNRS-UMR 6625, Campus de Beaulieu, Bâtiment 22, 35042 Rennes cedex, France.
nathalie.krell@univ-rennes1.fr

Received: 4 November 2015
Revised: 3 May 2016
Accepted: 11 May 2016

Abstract

We consider the class of Piecewise deterministic Markov processes (PDMP), whose state space is (0,∞), that possess an increasing deterministic motion and with a deterministic jump mechanism. 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” Escherichia coli cell. Having observed the PDMP until its nth jump, we construct a nonparametric estimator of the jump rate λ. Our main result is that for 𝓓 a compact subset of (0,∞), if λ is in the Hölder space 𝓗^s(𝓓), the squared-loss error of the estimator is asymptotically close to the speed of n^{− s/ (2s + 1)}. Simulations illustrate the behavior of our estimator.