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Nonparametric estimation of the derivatives of the stationarydensity for stationary processes

Published online by Cambridge University Press:  06 December 2012

Emeline Schmisser
Emeline Schmisser*
Affiliation:
UniversitéLille 1, Laboratoire Paul Painlevé, Cité Scientifique, 59655 Villeneuve d’Ascq, France.. emeline.schmisser@math.univ-lille1.fr
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Abstract

In this article, our aim is to estimate the successive derivatives of the stationarydensity f of a strictly stationary and β-mixing process (Xt)t≥0. This process is observed at discrete timest = 0,Δ,...,nΔ. Thesampling interval Δ can be fixed or small. We use a penalizedleast-square approach to compute adaptive estimators. If the derivativef(j) belongs to the Besov space \hbox{$\rond{B}_{2,\infty}^{\alpha}$}B2,∞α, then our estimator converges at rate (nΔ)−α/(2α+2j+1). Then we consider a diffusion with known diffusioncoefficient. We use the particular form of the stationary density to compute an adaptiveestimator of its first derivative f′. When the sampling intervalΔ tends to 0, and when the diffusion coefficient is known, theconvergence rate of our estimator is (nΔ)−α/(2α+1). When the diffusion coefficient is known, we alsoconstruct a quotient estimator of the drift for low-frequency data.

Keywords

Derivatives of the stationary densitydiffusion processesmixing processesnonparametric estimationstationary processes.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 33 - 69
Copyright
© EDP Sciences, SMAI, 2012

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