## Bernoulli

- Bernoulli
- Volume 25, Number 3 (2019), 1870-1900.

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

Masaaki Fukasawa and Tetsuya Takabatake

#### Abstract

This paper proposes feasible asymptotically efficient estimators for a certain class of Gaussian noises with self-similarity and stationarity properties, which includes the fractional Gaussian noises, 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. Further by Monte-Carlo experiments, we examine finite sample performances of our estimators. Finite sample modifications of the asymptotic variances of the estimators are also given, which exhibit almost perfect fits to the numerical results.

#### Article information

**Source**

Bernoulli, Volume 25, Number 3 (2019), 1870-1900.

**Dates**

Received: August 2017

Revised: March 2018

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1560326431

**Digital Object Identifier**

doi:10.3150/18-BEJ1039

**Mathematical Reviews number (MathSciNet)**

MR3961234

**Zentralblatt MATH identifier**

07066243

**Keywords**

asymptotic efficiency fractional Gaussian noises high frequency observations local asymptotic normality Whittle estimation

#### Citation

Fukasawa, Masaaki; Takabatake, Tetsuya. Asymptotically efficient estimators for self-similar stationary Gaussian noises under high frequency observations. Bernoulli 25 (2019), no. 3, 1870--1900. doi:10.3150/18-BEJ1039. https://projecteuclid.org/euclid.bj/1560326431

#### Supplemental materials

- Supplement to “Asymptotically efficient estimators for self-similar stationary Gaussian noises under high frequency observations”. We explain how to implement spectral densities of self-similar stationary Gaussian noises and their derivatives with respect to parameters for more detail. This procedure is applicable for all examples shown in Section 4 of the original article.Digital Object Identifier: doi:10.3150/18-BEJ1039SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.