## Bernoulli

• 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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 28 February 2014

https://projecteuclid.org/euclid.bj/1393594005

Digital Object Identifier
doi:10.3150/12-BEJ505

Mathematical Reviews number (MathSciNet)
MR3178517

Zentralblatt MATH identifier
06291821

#### Citation

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. https://projecteuclid.org/euclid.bj/1393594005

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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, ….