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Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes

Part of: Markov processes Nonparametric inference

Published online by Cambridge University Press:  30 January 2018

Takayuki Fujii
Takayuki Fujii*
Affiliation:
Osaka University
*
∗ Current address: Faculty of Economics, Shiga University, 1-1-1 Banba, Hikone, Shiga 522-8522, Japan. Email address: takayuki-fujii@biwako.shiga-u.ac.jp
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Abstract

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In this paper we study nonparametric estimation problems for a class of piecewise-deterministic Markov processes (PDMPs). Borovkov and Last (2008) proved a version of Rice's formula for PDMPs, which explains the relation between the stationary density and the level crossing intensity. From a statistical point of view, their result suggests a methodology for estimating the stationary density from observations of a sample path of PDMPs. First, we introduce the local time related to the level crossings and construct the local-time estimator for the stationary density, which is unbiased and uniformly consistent. Secondly, we investigate other estimation problems for the jump intensity and the conditional jump size distribution.

Keywords

Piecewise-deterministic Markov processlocal timestationary densityuniform consistencynonparametric estimation

MSC classification

Primary: 62G20: Asymptotic properties
Secondary: 60J55: Local time and additive functionals
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 931 - 942
Copyright
© Applied Probability Trust 

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