NONPARAMETRIC STATISTICAL INFERENCE FOR DRIFT VECTOR FIELDS OF MULTI-DIMENSIONAL DIFFUSIONS

• 成果类型: Article
• 署名作者: Nickl, Richard; Ray, Kolyan
• 署名单位: University of Cambridge; University of London; King's College London
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1851
• 发表日期: 2020
• 页码: 1383-1408
• 关键词: von mises theorems; Posterior Contraction Rates; bayesian inverse problems; ergodic diffusions; convergence-rates; distributions; EQUIVALENCE; functionals

摘要

The problem of determining a periodic Lipschitz vector field b = (b(1),..., b(d)) from an observed trajectory of the solution (X-t : 0 <= t <= T) of the multi-dimensional stochastic differential equation dX(t) = b(X-t) dt + dW(t), t >= 0, where W-t is a standard d-dimensional Brownian motion, is considered. Convergence rates of a penalised least squares estimator, which equals the maxi-mum a posteriori (MAP) estimate corresponding to a high-dimensional Gaus-sian product prior, are derived. These results are deduced from corresponding contraction rates for the associated posterior distributions. The rates obtained are optimal up to log-factors in L-2-loss in any dimension, and also for supre-mum norm loss when d <= 4. Further, when d <= 3, nonparametric Bernstein-von Mises theorems are proved for the posterior distributions of b. From this, we deduce functional central limit theorems for the implied estimators of the invariant measure mu(b). The limiting Gaussian process distributions have a covariance structure that is asymptotically optimal from an information-theoretic point of view.