Asymptotic equivalence of spectral density estimation and gaussian white noise

TL;DR

This paper proves that estimating the spectral density of a stationary Gaussian process is asymptotically equivalent to simpler Gaussian models, including a white noise model with log-spectral density, facilitating easier inference.

Contribution

It establishes the asymptotic equivalence of spectral density estimation to Gaussian scale regression and white noise models, simplifying complex spectral inference problems.

Findings

Asymptotic equivalence to Gaussian scale regression

Asymptotic equivalence to Gaussian white noise model

Connection to log-periodogram regression

Abstract

We consider the statistical experiment given by a sample of a stationary Gaussian process with an unknown smooth spectral density f. Asymptotic equivalence, in the sense of Le Cam's deficiency Delta-distance, to two Gaussian experiments with simpler structure is established. The first one is given by independent zero mean Gaussians with variance approximately the value of f in points of a uniform grid (nonparametric Gaussian scale regression). This approximation is closely related to well-known asymptotic independence results for the periodogram and corresponding inference methods. The second asymptotic equivalence is to a Gaussian white noise model where the drift function is the log-spectral density. This represents the step from a Gaussian scale model to a location model, and also has a counterpart in established inference methods, i.e. log-periodogram regression. The problem of simple explicit equivalence maps (Markov kernels), allowing to directly carry over inference, appears in this context but is not solved here.