The Annals of Statistics

Nonparametric estimation of scalar diffusions based on low frequency data

Emmanuel Gobet, Marc Hoffmann, and Markus Reiß

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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.

Article information

Ann. Statist., Volume 32, Number 5 (2004), 2223-2253.

First available in Project Euclid: 27 October 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G99: None of the above, but in this section 62M05: Markov processes: estimation 62M15: Spectral analysis

Diffusion processes nonparametric estimation discrete sampling low frequency data spectral approximation ill-posed problems


Gobet, Emmanuel; Hoffmann, Marc; Reiß, Markus. Nonparametric estimation of scalar diffusions based on low frequency data. Ann. Statist. 32 (2004), no. 5, 2223--2253. doi:10.1214/009053604000000797.

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