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

TL;DR

This paper develops efficient estimators for the scale parameter in Gaussian time series with long-range dependence and derives explicit Fisher information expressions, revealing a phase transition in asymptotic regimes based on spectral density behavior.

Contribution

It introduces explicit, asymptotically efficient estimators for the scale parameter and provides closed-form Fisher information formulas, highlighting a phase transition in asymptotic behavior.

Findings

Fisher information exhibits two distinct asymptotic regimes.

Estimates are explicitly computable and asymptotically efficient.

Phase transition depends on spectral density behavior at zero.

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.