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

## TL;DR

This paper introduces a nonparametric method to estimate the jump rate of a specific class of PDMPs, using an adaptive stationary density estimator and a quotient estimator, with theoretical risk bounds and simulation validation.

## Contribution

It develops a novel adaptive estimation procedure for the jump rate of PDMPs, achieving nearly minimax optimality with theoretical guarantees.

## Key findings

- Estimator of jump rate is nearly minimax optimal.
- Uniform risk bounds are established for the estimators.
- Simulations demonstrate the estimator's effectiveness.

## Abstract

In this paper, we consider a piecewise deterministic Markov process (PDMP), with known flow and deterministic transition measure, and unknown jump rate $\lambda$. To estimate nonparametrically the jump rate, we first construct an adaptive estimator of the stationary density, then we derive a quotient estimator $\hat{\lambda}_n$ of $\lambda$. We provide uniform bounds for the risk of these estimators, and prove that the estimator of the jump rate is nearly minimax (up to a $\ln^2(n)$ factor). Simulations illustrate the behavior of our estimator.