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.
Mots clés : derivatives of the stationary density, diffusion processes, mixing processes, nonparametric estimation, stationary processes
@article{PS_2013__17__33_0,
author = {Schmisser, Emeline},
title = {Nonparametric estimation of the derivatives of the stationary density for stationary processes},
journal = {ESAIM: Probability and Statistics},
pages = {33--69},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2011102},
mrnumber = {3002995},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2011102/}
}
TY - JOUR AU - Schmisser, Emeline TI - Nonparametric estimation of the derivatives of the stationary density for stationary processes JO - ESAIM: Probability and Statistics PY - 2013 SP - 33 EP - 69 VL - 17 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2011102/ DO - 10.1051/ps/2011102 LA - en ID - PS_2013__17__33_0 ER -
%0 Journal Article %A Schmisser, Emeline %T Nonparametric estimation of the derivatives of the stationary density for stationary processes %J ESAIM: Probability and Statistics %D 2013 %P 33-69 %V 17 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2011102/ %R 10.1051/ps/2011102 %G en %F PS_2013__17__33_0
Schmisser, Emeline. Nonparametric estimation of the derivatives of the stationary density for stationary processes. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 33-69. doi : 10.1051/ps/2011102. http://www.numdam.org/articles/10.1051/ps/2011102/
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