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

Abstract : Abstract In this article, our aim is to estimate the successive derivatives of the stationary density $f$ of a strictly stationary and $\beta$-mixing process$\left(X_{t}\right)_{t\geq0}$. This process is observed at discrete times $t=0,\Delta,\ldots,n\Delta$. The sampling interval $\Delta$ 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 $\mathcal{B}_{2,\infty}^{\alpha}$, then our estimator converges at rate $\left(n\Delta\right)^{-\alpha/(2\alpha+2j+1)}$. 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 $\Delta$ tends to 0, and when the diffusion coefficient is known, the convergence rate of our estimator is $\left(n\Delta\right)^{-\alpha/(2\alpha+1)}$. When the diffusion coefficient is known, we also construct a quotient estimator of the drift for low-frequency data.
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Article dans une revue
ESAIM: Probability and Statistics, EDP Sciences, 2013, 17, pp33-69. <10.1080/02331888.2011.591931>
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https://hal.archives-ouvertes.fr/hal-00507025
Contributeur : Emeline Schmisser <>
Soumis le : vendredi 30 juillet 2010 - 11:09:39
Dernière modification le : mardi 11 octobre 2016 - 11:59:02
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Emeline Schmisser. Nonparametric estimation of the derivatives of the stationary density for stationary processes. ESAIM: Probability and Statistics, EDP Sciences, 2013, 17, pp33-69. <10.1080/02331888.2011.591931>. <hal-00507025>

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