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May 2014 Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information
Till Sabel, Johannes Schmidt-Hieber
Bernoulli 20(2): 747-774 (May 2014). DOI: 10.3150/12-BEJ505

Abstract

Mimicking the maximum likelihood estimator, we construct first order Cramer–Rao efficient and explicitly computable estimators for the scale parameter $\sigma^{2}$ in the model $Z_{i,n}=\sigma n^{-\beta}X_{i}+Y_{i}$, $i=1,\ldots,n$, $\beta>0$ with independent, stationary Gaussian processes $(X_{i})_{i\in\mathbb{N}}$, $(Y_{i})_{i\in\mathbb{N}}$, and $(X_{i})_{i\in\mathbb{N}}$ exhibits possibly long-range dependence. In a second part, closed-form expressions for the asymptotic behavior of the corresponding Fisher information are derived. Our main finding is that depending on the behavior of the spectral densities at zero, the Fisher information has asymptotically two different scaling regimes, which are separated by a sharp phase transition. The most prominent example included in our analysis is the Fisher information for the scaling factor of a high-frequency sample of fractional Brownian motion under additive noise.

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Till Sabel. Johannes Schmidt-Hieber. "Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information." Bernoulli 20 (2) 747 - 774, May 2014. https://doi.org/10.3150/12-BEJ505

Information

Published: May 2014
First available in Project Euclid: 28 February 2014

zbMATH: 06291821
MathSciNet: MR3178517
Digital Object Identifier: 10.3150/12-BEJ505

Rights: Copyright © 2014 Bernoulli Society for Mathematical Statistics and Probability

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Vol.20 • No. 2 • May 2014
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