Electronic Journal of Statistics

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

Romain Azaïs and Aurélie Muller-Gueudin

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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.

Article information

Source
Electron. J. Statist., Volume 10, Number 2 (2016), 3648-3692.

Dates
Received: October 2015
First available in Project Euclid: 3 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1480734074

Digital Object Identifier
doi:10.1214/16-EJS1207

Mathematical Reviews number (MathSciNet)
MR3579198

Zentralblatt MATH identifier
1353.62091

Subjects
Primary: 62M05: Markov processes: estimation 62G20: Asymptotic properties 60J25: Continuous-time Markov processes on general state spaces

Keywords
Cross-validation jump rate kernel method nonparametric estimation piecewise-deterministic Markov process

Citation

Azaïs, Romain; Muller-Gueudin, Aurélie. Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes. Electron. J. Statist. 10 (2016), no. 2, 3648--3692. doi:10.1214/16-EJS1207. https://projecteuclid.org/euclid.ejs/1480734074


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