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October 2004 Nonparametric estimation of scalar diffusions based on low frequency data
Emmanuel Gobet, Marc Hoffmann, Markus Reiß
Ann. Statist. 32(5): 2223-2253 (October 2004). DOI: 10.1214/009053604000000797


We study the problem of estimating the coefficients of a diffusion (Xt,t0); the estimation is based on discrete data XnΔ,n=0,1,,N. The sampling frequency Δ1 is constant, and asymptotics are taken as the number N 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-error loss 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.

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 (XnΔ,n=0,1,,N) in a suitable Sobolev norm, together with an estimation of its invariant density.


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Emmanuel Gobet. Marc Hoffmann. Markus Reiß. "Nonparametric estimation of scalar diffusions based on low frequency data." Ann. Statist. 32 (5) 2223 - 2253, October 2004.


Published: October 2004
First available in Project Euclid: 27 October 2004

zbMATH: 1056.62091
MathSciNet: MR2102509
Digital Object Identifier: 10.1214/009053604000000797

Primary: 62G99 , 62M05 , 62M15

Keywords: Diffusion processes , discrete sampling , Ill-posed problems , low frequency data , nonparametric estimation , spectral approximation

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 5 • October 2004
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