Asymptotic equivalence of spectral density estimation and Gaussian white noise
Georgi K. Golubev, Michael Nussbaum, Harrison H. Zhou
The Annals of Statistics · 2010 · Volume 38 · Issue 1
Abstract
We consider the statistical experiment given by a sample y(1), …, y(n) of a stationary Gaussian process with an unknown smooth spectral density f. Asymptotic equivalence, in the sense of Le Cam’s deficiency Δ-distance, to two Gaussian experiments with simpler structure is established. The first one is given by independent zero mean Gaussians with variance approximately f(ωi), where ωi is a uniform grid of points in (−π, π) (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, that is, 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.
Keywords
62G07, 62G20, asymptotic equivalence, Le Cam distance, log-periodogram regression, nonparametric Gaussian scale model, signal in Gaussian white noise, Sobolev classes, Spectral density, stationary Gaussian process, Whittle likelihood
Enable JavaScript for citation tools and additional paper interactions.