Translator Disclaimer
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

Abstract

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.

Citation

Download Citation

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

Information

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

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

Subjects:
Primary: 62G99, 62M05, 62M15

Rights: Copyright © 2004 Institute of Mathematical Statistics

JOURNAL ARTICLE
31 PAGES


SHARE
Vol.32 • No. 5 • October 2004
Back to Top