Asymptotically efficient estimators for self-similar stationary Gaussian noises under high frequency observations

TL;DR

This paper develops feasible, asymptotically efficient estimators for self-similar stationary Gaussian noises, including fractional Gaussian noise, under high frequency data, addressing the challenges posed by parameter singularities.

Contribution

It extends the Whittle estimation method to high frequency observations and achieves asymptotic efficiency for estimating Hurst and diffusion parameters.

Findings

Estimators are asymptotically efficient in Fisher's sense.

The method handles the singularity of the Fisher information matrix.

Performance depends on known or unknown parameters.

Abstract

This paper proposes feasible asymptotically efficient estimators for a certain class of Gaussian noises with self-similar and stationary properties, which includes the fractional Gaussian noise, under high frequency observations. In this setting, the optimal rate of estimation depends on whether either the Hurst or diffusion parameters is known or not. This is due to the singularity of the asymptotic Fisher information matrix for simultaneous estimation of the above two parameters. One of our key ideas is to extend the Whittle estimation method to the situation of high frequency observations. We show that our estimators are asymptotically efficient in Fisher's sense.