Nonparametric estimation of scalar diffusions based on low frequency data

Details Nonparametric estimation of scalar diffusions based on low frequency data Nonparametric estimation of scalar diffusions based on low frequency data

2005-03-29

arxiv.org/abs/math/0503680v1

Annals of Statistics 2004, Vol. 32, No. 5, 2223-2253

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053604000000797 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053604000000797

We study the problem of estimating the coefficients of a diffusion (X_t,t\geq 0); the estimation is based on discrete data X_{n\Delta},n=0,1,...,N. The sampling frequency \Delta^{-1} 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 (X_{n\Delta},n=0,1,...,N) in a suitable Sobolev norm, together with an estimation of its invariant density.

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