Second Moment Estimator for An AR(1) Model Driven by A Long Memory Gaussian Noise

TL;DR

This paper introduces a second moment estimator for an AR(1) model driven by long memory Gaussian noise, proving its consistency and asymptotic distribution, with bounds when the distribution is Gaussian.

Contribution

The paper proposes a novel second moment estimator for AR(1) models with long memory Gaussian noise, establishing its strong consistency and asymptotic properties.

Findings

Estimator is strongly consistent.

Asymptotic distribution derived, Gaussian case includes Berry-Esséen bounds.

Applicable to fractional Gaussian noise and fractional ARIMA models.

Abstract

In this paper, we consider an inference problem for the first order autoregressive process driven by a long memory stationary Gaussian process. Suppose that the covariance function of the noise can be expressed as $\abs{k}^{2H-2}$ times a function slowly varying at infinity. The fractional Gaussian noise and the fractional ARIMA model and some others Gaussian noise are special examples that satisfy this assumption. We propose a second moment estimator and prove the strong consistency and give the asymptotic distribution. Moreover, when the limit distribution is Gaussian, we give the upper Berry-Ess\'een bound by means of Fourth moment theorem.