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  1. Nonparametric estimation of scalar diffusions based on low frequency data is ill-posed
    Author: Gobet, Emmanuel; Hoffmann, Marc; Reiß, Markus
    Published: 2002
    Publisher:  Humboldt-Universität, Berlin

    We study the problem of estimating the coefficients of a diffusion (Xl, t 2:: 0); the estimation is based on discrete data Xn . . n = 0, 1, ... ,N. The sampling frequency delta t is constant , and asymptotics arc taken at the number of observations... more

    Location:
    Staats- und Universitätsbibliothek Bremen
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    Location:
    ZBW - Leibniz-Informationszentrum Wirtschaft, Standort Kiel
    Signature:
    DS 20 (2002,57)
    Inter-library loan:
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    We study the problem of estimating the coefficients of a diffusion (Xl, t 2:: 0); the estimation is based on discrete data Xn . . n = 0, 1, ... ,N. The sampling frequency delta t is constant , and asymptotics arc taken at the number of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient - the volatility - and the drift in a nonparametric setting is ill-posed: The minimax rates of convergence for Sobolev constraints and squared-crror lOBS coincide with that of a respectively first and second order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. An important consequence of this result is that we can characterize quantitatively the difference between the estimation of a diffusion in the low frequency sampling case and inference problems in other related frameworks: nonparametric estimation of a diffusion based on continuous or high frequency data: but also parametric estimation for fixed delta, in which case root-N-consistent estimators usually exist. Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue-eigenfunction pair of the transition operator of the discrete time Markov chain (X/lA, n = 0,1, ... ,N) in a suitable Sobolev norm: together with an estimation of its invariant density. -- Diffusion processes ; Nonparametric estimation ; Discrete Sarnpling ; Low frequency data ; Spectral approximation ; Ill-posed problems

     
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    Content information
    Source: Union catalogues
    Language: English
    Media type: Book
    Format: Online
    Other identifier:
    hdl: 10419/65321
    Series: Discussion papers of interdisciplinary research project 373 ; 2002,57
    Scope: Online-Ressource (PDF-Datei: 38 S., 1,75 MB)
  2. Nonparametric estimation of scalar diffusions based on low frequency data is ill-posed
    Author: Gobet, Emmanuel; Hoffmann, Marc; Reiß, Markus
    Published: 2002
    Publisher:  Sonderforschungsbereich 373, Berlin

    Location:
    ZBW - Leibniz-Informationszentrum Wirtschaft, Standort Kiel
    Signature:
    W 1190 (2002.57)
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    Source: Union catalogues
    Language: English
    Media type: Book
    Format: Print
    Series: Discussion paper / Humboldt-Universität zu Berlin, Sonderforschungsbereich 373, Quantifikation und Simulation Ökonomischer Prozesse ; 2002,57
    Subjects: Nichtparametrisches Verfahren; Schätztheorie; Markov-Kette; Theorie
    Scope: 38 S
    Notes: