Local estimation of transition rates of jump processes through discretization

TL;DR

This paper develops a nonparametric Poisson regression approach for estimating transition rates in Markov and semi-Markov jump processes, demonstrating asymptotic normality without structural assumptions.

Contribution

It introduces a flexible discretization method allowing variable interval lengths and proves asymptotic normality under minimal assumptions.

Findings

Asymptotic normality of rate estimators established for both models.

Method validated on simulated data and real datasets.

No structural assumptions needed on true intensities.

Abstract

We investigate the Poisson regression method for Markov and semi-Markov jump processes from a nonparametric angle, allowing the lengths of the time and duration intervals in the partition to vary with the number of observations. Imposing no structural assumptions on the true intensities, we obtain asymptotic normality of the occurence/exposure rates under appropriate shrinking conditions on the partition lengths. We derive asymptotic normality results for both Markov and semi-Markov models using only classical central limit theorems and elementary results for counting processes. All results are illustrated on both simulated and real data.