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Some problems of nonparametric estimation by observations of ergodic diffusion process

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  • Kutoyants, Yu. A.

Abstract

We consider the problems of the density and distribution function estimation by the observations of diffusion process with ergodic properties. In every problem we first propose a minimax bound on the risk of any estimator and then study the asymptotic behavior of several estimators. It is shown that the empiric distribution function is asymptotically normal and asymptotically efficient (in the minimax sense) estimator of the distribution function. In the density estimation problem, we describe the asymptotic behavior of a kernel-type estimator and one another (unbiased) estimator. Both of them are [radical sign]T-consistent, asymptotically normal and asymptotically efficient.

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  • 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.
    • Handle: RePEc:eee:stapro:v:32:y:1997:i:3:p:311-320
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    References listed on IDEAS

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    1. Castellana, J. V. & Leadbetter, M. R., 1986. "On smoothed probability density estimation for stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 21(2), pages 179-193, February.
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    1. Guillou, Armelle & Merlevède, Florence, 2001. "Estimation of the Asymptotic Variance of Kernel Density Estimators for Continuous Time Processes," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 114-137, October.
    2. 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.
    3. Bosq, Denis & Merlevède, Florence & Peligrad, Magda, 1999. "Asymptotic Normality for Density Kernel Estimators in Discrete and Continuous Time," Journal of Multivariate Analysis, Elsevier, vol. 68(1), pages 78-95, January.

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