Parametric rates of nonparametric estimators and predictors for continuous time processes

• 成果类型: Article
• 署名作者: Bosq, D
• 署名单位: Sorbonne Universite
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1069362734
• 发表日期: 1997
• 页码: 982-1000
• 关键词: random-variables; series

摘要

We show that local irregularity of observed sample paths provides additional information which allows nonparametric estimators and predictors for continuous time processes to reach parametric rates in mean square as well as in a.s. uniform convergence. For example, we prove that under suitable conditions the kernel density estimator f(T) associated with the observed sample path (X-t, 0 less than or equal to t less than or equal to T) satisfies [GRAPHICS] where f denotes the unknown marginal density of the stationary process (X-t)and where ln(k) denotes the kth iterated logarithm. The proof uses a special Borel-Cantelli lemma for continuous time processes together with a sharp large deviation inequality. Furthermore the parametric rate obtained in (1) is preserved by using a suitable sampling scheme.