Nonparametric estimation of a renewal reward process from discrete data

TL;DR

This paper introduces an adaptive wavelet-based method for nonparametrically estimating the jump density of a renewal reward process from discrete data, achieving minimax convergence rates despite dependent, non-stationary increments.

Contribution

It develops a novel wavelet threshold estimator for renewal reward processes, handling the non-closed form inverse of the compounding operator with a fixed point approach.

Findings

Achieves minimax rates of convergence for the estimator.

Handles dependent, non-stationary increments in the process.

Provides an adaptive estimation procedure for discrete observations.

Abstract

We study the nonparametric estimation of the jump density of a renewal reward process from one discretely observed sample path over [0,T]. We consider the regime when the sampling rate goes to 0. The main difficulty is that a renewal reward process is not a Levy process: the increments are non stationary and dependent. We propose an adaptive wavelet threshold density estimator and study its performance for the Lp loss over Besov spaces. We achieve minimax rates of convergence for sampling rates that vanish with T at polynomial rate. In the same spirit as Buchmann and Gr\"ubel (2003) and Duval (2012), the estimation procedure is based on the inversion of the compounding operator. The inverse has no closed form expression and is approached with a fixed point technique.