Nonparametric estimation of the conditional distribution of the inter-jumping times for piecewise-deterministic Markov processes

TL;DR

This paper introduces a nonparametric approach to estimate the conditional distribution of inter-jumping times in piecewise-deterministic Markov processes, requiring only a single long-term observation and demonstrating consistency and effectiveness through simulations.

Contribution

It develops a novel nonparametric estimation method based on a generalized Aalen's model, applicable with minimal observational data and proven to be uniformly consistent.

Findings

Estimator is uniformly consistent under certain conditions

Method performs well in simulation studies

Requires only one long-term observation

Abstract

This paper presents a nonparametric method for estimating the conditional density associated to the jump rate of a piecewise-deterministic Markov process. In our framework, the estimation needs only one observation of the process within a long time interval. Our method relies on a generalization of Aalen's multiplicative intensity model. We prove the uniform consistency of our estimator, under some reasonable assumptions related to the primitive characteristics of the process. A simulation example illustrates the behavior of our estimator.