Nonparametric estimation of the jump rate for non-homogeneous marked renewal processes

TL;DR

This paper introduces a nonparametric method to estimate jump and cumulative rates of non-homogeneous marked renewal processes using a single long-term observation, extending Aalen's multiplicative intensity model.

Contribution

It develops consistent estimators for jump and cumulative rates in a general setting, broadening the applicability of renewal process analysis.

Findings

Estimates are consistent under ergodicity assumptions.

Method is applicable with only one long observation.

Numerical example demonstrates practical effectiveness.

Abstract

This paper is devoted to the nonparametric estimation of the jump rate and the cumulative rate for a general class of non-homogeneous marked renewal processes, defined on a separable metric space. In our framework, the estimation needs only one observation of the process within a long time. Our approach is based on a generalization of the multiplicative intensity model, introduced by Aalen in the seventies. We provide consistent estimators of these two functions, under some assumptions related to the ergodicity of an embedded chain and the characteristics of the process. The paper is illustrated by a numerical example.