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

Emeline Schmisser

ESAIM: Probability and Statistics (2013)

  • Volume: 17, page 33-69
  • ISSN: 1292-8100

Abstract

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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 B 2 , ∞ α B 2 , ∞ α , 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.

How to cite

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Schmisser, Emeline. "Nonparametric estimation of the derivatives of the stationary density for stationary processes." ESAIM: Probability and Statistics 17 (2013): 33-69. <http://eudml.org/doc/274349>.

@article{Schmisser2013,
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 (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 $\{B\}_\{2,\infty \}^\{\alpha \}$ B 2 , ∞ α , 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.},
author = {Schmisser, Emeline},
journal = {ESAIM: Probability and Statistics},
keywords = {derivatives of the stationary density; diffusion processes; mixing processes; nonparametric estimation; stationary processes},
language = {eng},
pages = {33-69},
publisher = {EDP-Sciences},
title = {Nonparametric estimation of the derivatives of the stationary density for stationary processes},
url = {http://eudml.org/doc/274349},
volume = {17},
year = {2013},
}

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
PB - EDP-Sciences
VL - 17
SP - 33
EP - 69
AB - 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 ${B}_{2,\infty }^{\alpha }$ B 2 , ∞ α , 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.
LA - eng
KW - derivatives of the stationary density; diffusion processes; mixing processes; nonparametric estimation; stationary processes
UR - http://eudml.org/doc/274349
ER -

References

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