Local estimation of transition rates of jump processes through discretization

2026-05-05

AI Summary

This study addresses the problem of local nonparametric estimation of transition intensities for Markov and semi-Markov jump processes without imposing structural assumptions on the underlying intensity functions. The authors propose a Poisson regression approach based on adaptive partitioning of time and sojourn duration, enabling localized modeling of occurrence/exposure rates through data-driven interval refinement. Relying solely on fundamental properties of counting processes and the central limit theorem, they establish the asymptotic normality of the proposed estimator without requiring prior assumptions such as smoothness or parametric forms for the true intensity functions. Theoretical analysis demonstrates that, under appropriate conditions on partition shrinkage, the estimator enjoys favorable asymptotic properties. Numerical simulations and empirical experiments further corroborate the method’s effectiveness and robustness.

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.

Problem

• jump processes
• transition rates
• nonparametric estimation
• asymptotic normality
• Poisson regression

Innovation

• nonparametric estimation
• jump processes
• asymptotic normality
• Poisson regression
• adaptive partitioning