Nonparametric estimation of scalar diffusions based on low frequency data

• 成果类型: Article
• 署名作者: Gobet, E; Hoffmann, M; Reiss, M
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique; ENSTA Paris; Universite Paris Cite; Sorbonne Universite; Humboldt University of Berlin
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053604000000797
• 发表日期: 2004
• 页码: 2223-2253
• 关键词: model

摘要

We study the problem of estimating the coefficients of a diffusion (X-t, t greater than or equal to 0); the estimation is based on discrete data X-nDelta, 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-nDelta, n = 0, 1,..., N) in a suitable Sobolev norm, together with an estimation of its invariant density.