Nonparametric estimation of the derivatives of the stationary density for stationary processes

UniversitéLille 1, Laboratoire Paul Painlevé, Cité Scientifique, 59655 Villeneuve d’Ascq, France.
emeline.schmisser@math.univ-lille1.fr

Received: 13 July 2010

Abstract

In this article, our aim is to estimate the successive derivatives of the stationary density f of a strictly stationary and β-mixing process (X_t)_t≥0. This process is observed at discrete times t = 0,Δ,...,nΔ. The sampling interval Δ can be fixed or small. We use a penalized least-square approach to compute adaptive estimators. If the derivative f^(j) belongs to the Besov space , then our estimator converges at rate (nΔ)^{−α/(2α+2j+1)}. Then we consider a diffusion with known diffusion coefficient. We use the particular form of the stationary density to compute an adaptive estimator of its first derivative f′. When the sampling interval Δ tends to 0, and when the diffusion coefficient is known, the convergence rate of our estimator is (nΔ)^{−α/(2α+1)}. When the diffusion coefficient is known, we also construct a quotient estimator of the drift for low-frequency data.