Nonparametric estimation of a renewal reward process from discrete data

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 regimes where the sampling rate Δ = Δ _T → 0 as T → ∞. We propose an adaptive wavelet threshold density estimator and study its performance for the L _p loss, p ≥ 1, over Besov spaces. We achieve minimax rates of convergence for sampling rates Δ _T that vanish with T at arbitrary 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.