Berry-Esséen bounds for parameter estimation of general Gaussian processes

Abstract

We study rates of convergence in central limit theorems for the partial sum of squares of general Gaussian sequences, using tools from analysis on Wiener space. No assumption of stationarity, asymptotically or otherwise, is made. The main theoretical tool is the so-called Optimal Fourth Moment Theorem [19], which provides a sharp quantitative estimate of the total variation distance on Wiener chaos to the normal law. The only assumptions made on the sequence are the existence of an asymptotic variance, that a least-squares-type estimator for this variance parameter has a bias and a variance which can be controlled, and that the sequence's auto-correlation function, which may exhibit long memory, has a no-worse memory than that of fractional Brownian motion with Hurst parameter H < 3/4. Our main result is explicit, exhibiting the trade-off between bias, variance, and memory. We apply our result to study drift parameter estimation problems for subfractional Ornstein-Uhlenbeck and bifractional Ornstein-Uhlenbeck processes with fixed-time-step observations. These are processes which fail to be stationary or self-similar, but for which detailed calculations result in explicit formulas for the estimators' asymptotic normality.

AR(1) processes driven by second-chaos white noise: Berry–Esséen bounds for quadratic variation and parameter estimation

“…Following an initial push in [51], most of the recent papers mentioned above, and recent references therein, state an explicit effort to work with discretely observed processes. At least in the increasing-horizon case, the papers [18] and [13] had the merit of pointing out that many of the discretization techniques used to pass from continuous-path to discrete-observation based estimators, were inefficient, and it is preferable to work directly from the statistics of the discretely observed process. Our paper picks up this thread, and introduces a new direction of research which, to our knowledge, has not been approached by any authors: can the asymptotic normality of quadratic variations and related estimators, including very precise results on speeds of convergence, be obtained when the driving noise is not Gaussian?…”

Asymptotic properties for the parameter estimation in Ornstein-Uhlenbeck process with discrete observations

“…where z α/2 is the upper α/2-quantile of standard normal distribution. Here, the quality of approximation for above confidence interval can be an assessed by the Berry-Esseen bound of estimatorθ n,Δ ([1], [11], [14], [15]).…”

“…Moreover, there are also some research works on maximum likelihood estimator of θ in continuous observation case. For stationary case, one can refer to ([4], [6], [16]) for the large deviations, and ( [17], [18], [20], [26]) for Cramértype moderate deviations, and ( [1], [11], [14], [15]) for the Berry-Esseen bounds. In explosive case, Bercu et al ([5]), Bercu and Richou ([7]) obtained the large deviations for the maximum likelihood estimator of θ, while Jiang and Zhang ( [27]) considered the Cramér-type moderate deviations.…”

Berry-Esseen bounds of second moment estimators for Gaussian processes observed at high frequency

“…The method of moments is more computationally tractable especially when one considers discrete estimators. Some recent studies include [11,17,21] where the mesh in time Δ n = 1 in (1.2), which is akin to the discretization of a least-squares method for fractional Gaussian processes using fixed-time-step observations. See also [8] for an application of the second moment method to an AR(1) model.…”

“…On the other hand, the consistency and speed of convergence in the TV and Wasserstein norms for the estimator (1.1) of the drift parameter in infinite dimensional linear stochastic equations driven by a fBm are studied by [26]. • The case of discrete-time observations for ergodic-type Gaussian processes, using (1.2): In the case when the mesh in time Δ n = 1, the consistency and speed of convergence in the TV and Wasserstein distance for the estimator (1.2) of the limiting variance of of general Gaussian sequences were recently developed in the papers [17,11]. Also, the drift parameter in linear stochastic evolution equation driven by a fBm is considered in [26].…”