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

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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


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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.