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

ESAIM: Probability and Statistics (2013)

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

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topSchmisser, 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 -

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