  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.
