• Bernoulli
  • Volume 20, Number 2 (2014), 747-774.

Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information

Till Sabel and Johannes Schmidt-Hieber

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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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Bernoulli, Volume 20, Number 2 (2014), 747-774.

First available in Project Euclid: 28 February 2014

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efficient estimation fractional Brownian motion Fisher information regular variation slowly varying function spectral density


Sabel, Till; Schmidt-Hieber, Johannes. Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information. Bernoulli 20 (2014), no. 2, 747--774. doi:10.3150/12-BEJ505.

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Supplemental materials

  • Supplementary material: Supplement to “Asymptotically efficient estimation of a scale parameter in Gaussian time series and closed-form expressions for the Fisher information”. In the supplement, we provide the proofs of Theorem 3 and Corollary 1 along with some technical propositions and lemmas denoted by B.1, B.2, …, C.1, C.2, ….