Uniform Convergence Rates of Kernel-Based Nonparametric Estimators for Continuous Time Diffusion Processes: A Damping Function Approach

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 adamping 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.

Estimation of Stochastic Volatility Models by Nonparametric Filtering

“…This result is often needed when one considers nonparametric estimators for continuous-time processes under long span asymptotics, and should be of independent interest; see Kanaya (2014) We restrict the set of feasible bandwidth sequences that can be used to estimate the trajectory of σ 2 t :…”

“…Unfortunately, the martingale property is not readily preserved under truncation and so the precise argument is quite involved and leads to the additional, nonstandard term. Exponential inequalities combined with truncation are a standard tool for deriving uniform rates; see, e.g., Hansen (2008), Kristensen (2009), Gao, Kanaya, Li andTjøstheim (2015), and Kanaya (2014). However, these papers assume a mixing (or an i.i.d.)…”

CONVERGENCE RATES OF SUMS OF α-MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES

“…0 and T ! 1, which is discussed and veri…ed in Kanaya (2016), where the proof of (50) is based on a new result on the global modulus of continuity of Brownian motions (also developed in Kanaya, 2016), as well as so-called time-change arguments. Given (50), we can check (49) since the OU process satis…es maxf1; sup 0 s<T j Z (Z s )jg 1 + Z = O (1) and sup 0 s<T ja Z (Z s )j j Z m z j + O p (sup 0 s<T jZ s j) = O p ( p log T ), where the latter follows from the result on extremal processes (see, e.g., Borkovec and Klüpperlberg, 1998; Jeong and Park, 2014, and references therein).…”

“…under certain conditions (see discussions on the global modulus of continuity in Kanaya, 2016). We can then let n = p log (1= ) in (22), which may be satis…ed by Example 3.…”

“…Finally, the moment conditions imposed in B4 and/or B5 may be too strong for some classes of processes. Even in such cases, we may be able to derive the convergence rate of the estimators by using the damping function approach considered in Kanaya (2016)…”