UNIFORM CONVERGENCE RATES OF KERNEL-BASED NONPARAMETRIC ESTIMATORS FOR CONTINUOUS TIME DIFFUSION PROCESSES: A DAMPING FUNCTION APPROACH. (14th July 2016)

• Record Type: Journal Article
• Authors: Kanaya, Shin
• Is Part Of: Econometric theory. Volume 33:Number 4(2017)
• Page Start: 874
• Page End: 914
• Publication Date: 2016-07-14
• Subjects: Econometrics -- Periodicals; 330.01519505
• DOI: 10.1017/S0266466616000219
• Languages: English
• ISSNs: 0266-4666

Abstract

Abstract : In this paper, we derive uniform convergence rates of nonparametric estimators for continuous time diffusion processes. In particular, we consider kernel-based estimators of the Nadaraya–Watson type, introducing a new technical device called a damping function . This device allows us to derive sharp uniform rates over an infinite interval with minimal requirements on the processes: The existence of the moment of any order is not required and the boundedness of relevant functions can be significantly relaxed. Restrictions on kernel functions are also minimal: We allow for kernels with discontinuity, unbounded support, and slowly decaying tails. Our proofs proceed by using the covering-number technique from empirical process theory and exploiting the mixing and martingale properties of the processes. We also present new results on the path-continuity property of Brownian motions and diffusion processes over an infinite time horizon. These path-continuity results, which should also be of some independent interest, are used to control discretization biases of the nonparametric estimators. The obtained convergence results are useful for non/semiparametric estimation and testing problems of diffusion processes.