A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process

TL;DR

This paper introduces a recursive nonparametric estimator for the transition density of a non-stationary piecewise-deterministic Markov process, using only a single long-term observation, with proven consistency and asymptotic normality.

Contribution

It proposes a novel recursive estimation method for the transition kernel of PDM processes based on invariant measure estimation, applicable from minimal observational data.

Findings

Estimator is consistent under general conditions

Central limit theorem established for the estimator

Simulation confirms good asymptotic properties

Abstract

In this paper, we investigate a nonparametric approach to provide a recursive estimator of the transition density of a non-stationary piecewise-deterministic Markov process, from only one observation of the path within a long time. In this framework, we do not observe a Markov chain with transition kernel of interest. Fortunately, one may write the transition density of interest as the ratio of the invariant distributions of two embedded chains of the process. Our method consists in estimating these invariant measures. We state a result of consistency and a central limit theorem under some general assumptions about the main features of the process. A simulation study illustrates the well asymptotic behavior of our estimator.