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Super optimal rates for nonparametric density estimation via projection estimators

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  • Comte, F.
  • Merlevède, F.

Abstract

In this paper, we study the problem of the nonparametric estimation of the marginal density f of a class of continuous time processes. To this aim, we use a projection estimator and deal with the integrated mean square risk. Under Castellana and Leadbetter's condition (Stoch. Proc. Appl. 21 (1986) 179), we show that our estimator reaches a parametric rate of convergence and coincides with the projection of the local time estimator. Discussions about the optimality of this condition are provided. We also deal with sampling schemes and the corresponding discretized processes.

Suggested Citation

  • Comte, F. & Merlevède, F., 2005. "Super optimal rates for nonparametric density estimation via projection estimators," Stochastic Processes and their Applications, Elsevier, vol. 115(5), pages 797-826, May.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:5:p:797-826
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    References listed on IDEAS

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    1. Veretennikov, A. Yu., 1997. "On polynomial mixing bounds for stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 115-127, October.
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    5. Yu. Kutoyants, 1998. "Efficient Density Estimation for Ergodic Diffusion Processes," Statistical Inference for Stochastic Processes, Springer, vol. 1(2), pages 131-155, May.
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    8. D. Blanke & B. Pumo, 2003. "Optimal sampling for density estimation in continuous time," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(1), pages 1-23, January.
    9. Kutoyants, Yu. A., 1997. "Some problems of nonparametric estimation by observations of ergodic diffusion process," Statistics & Probability Letters, Elsevier, vol. 32(3), pages 311-320, March.
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    Cited by:

    1. Comte, Fabienne & Prieur, Clémentine & Samson, Adeline, 2017. "Adaptive estimation for stochastic damping Hamiltonian systems under partial observation," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3689-3718.
    2. Karine Bertin & Nicolas Klutchnikoff & Fabien Panloup & Maylis Varvenne, 2020. "Adaptive estimation of the stationary density of a stochastic differential equation driven by a fractional Brownian motion," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 271-300, July.

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