Nonparametric estimation of a renewal reward process from discrete data

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 Grübel (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.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1203.3135

References in corpus (5)

• Diffusion of Earthquake Aftershock Epicenters, Omori's Law and Generalized Continuous-Time Random Walk Models
• A kernel type nonparametric density estimator for decompounding
• Probabilistic approach to a proliferation and migration dichotomy in the tumor cell invasion
• Risk bounds for the non-parametric estimation of Lévy processes
• A kinetic equation for linear fractional stable motion with applications to space plasma physics