Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes

TL;DR

This paper develops and analyzes a class of nonparametric kernel estimators for the jump rate in piecewise-deterministic Markov processes, focusing on their consistency, asymptotic normality, and optimal selection methods.

Contribution

It introduces a new class of recursive kernel estimators for the jump rate and proposes a method to select the optimal estimator based on variance minimization.

Findings

Proved strong pointwise consistency of the estimators.

Established asymptotic normality of the estimators.

Suggested cross-validation for bandwidth selection.

Abstract

A piecewise-deterministic Markov process is a stochastic process whose behavior is governed by an ordinary differential equation punctuated by random jumps occurring at random times. We focus on the nonparametric estimation problem of the jump rate for such a stochastic model observed within a long time interval under an ergodicity condition. We introduce an uncountable class (indexed by the deterministic flow) of recursive kernel estimates of the jump rate and we establish their strong pointwise consistency as well as their asymptotic normality. We propose to choose among this class the estimator with the minimal variance, which is unfortunately unknown and thus remains to be estimated. We also discuss the choice of the bandwidth parameters by cross-validation methods.