Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

New Berry-Esseen bounds for parameter estimation of Gaussian processes observed at high frequency

Khalifa Es-Sebaiy, Yong Chen

Abstract

The purpose of this paper is to estimate the limiting variance of asymptotically stationary Gaussian processes observed at high frequency, using the second moment estimator (SME). We study rates of convergence of the central limit theorem for the SME in terms of the total variation, Kolmogorov and Wasserstein distances, using some novel techniques and sharp estimates for cumulants. We apply our approach to provide Berry-Esseen bounds in Kolmogorov and Wasserstein distances for estimators of the drift parameters of Gaussian Ornstein-Uhlenbeck processes. Moreover, we prove that most of our estimates are strictly sharper than the ones obtained in the existing literature.

New Berry-Esseen bounds for parameter estimation of Gaussian processes observed at high frequency

Abstract

The purpose of this paper is to estimate the limiting variance of asymptotically stationary Gaussian processes observed at high frequency, using the second moment estimator (SME). We study rates of convergence of the central limit theorem for the SME in terms of the total variation, Kolmogorov and Wasserstein distances, using some novel techniques and sharp estimates for cumulants. We apply our approach to provide Berry-Esseen bounds in Kolmogorov and Wasserstein distances for estimators of the drift parameters of Gaussian Ornstein-Uhlenbeck processes. Moreover, we prove that most of our estimates are strictly sharper than the ones obtained in the existing literature.
Paper Structure (8 sections, 12 theorems, 138 equations)

This paper contains 8 sections, 12 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Elements of Malliavin calculus on Wiener space
  3. Main results
  4. Berry-Esseen bounds for a second moment estimator of $\mathbf{\mathbb{E}(Z_0^2)}$
  5. Berry-Esseen bounds for an estimator of $\mathbf{f\left(\mathbb{E}(Z_0^2)\right)}$
  6. Applications to Gaussian Ornstein-Uhlenbeck processes
  7. Fractional Ornstein-Uhlenbeck process of the first kind
  8. Fractional Ornstein-Uhlenbeck process of the second kind

Key Result

Lemma 3.1

Fix an integer $M \geq 2.$ We have where $\mathbf{k}=\left(k_{1}, \ldots, k_{M}\right)$ and $\mathbf{v} \in \mathbb{R}^{M}$ is a fixed vector whose components are 1 or -1.

Theorems & Definitions (26)

  • Lemma 3.1: NZ
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • ...and 16 more