Nonparametric estimation of the derivatives of the stationary density for stationary processes
UniversitéLille 1, Laboratoire Paul Painlevé, Cité Scientifique, 59655
Villeneuve d’Ascq, France.
In this article, our aim is to estimate the successive derivatives of the stationary density f of a strictly stationary and β-mixing process (Xt)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.
Mathematics Subject Classification: 62G05 / 60G10
Key words: Derivatives of the stationary density / diffusion processes / mixing processes / nonparametric estimation / stationary processes.
© EDP Sciences, SMAI, 2012