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February 2010 Asymptotic equivalence of spectral density estimation and Gaussian white noise
Georgi K. Golubev, Michael Nussbaum, Harrison H. Zhou
Ann. Statist. 38(1): 181-214 (February 2010). DOI: 10.1214/09-AOS705

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.

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Georgi K. Golubev. Michael Nussbaum. Harrison H. Zhou. "Asymptotic equivalence of spectral density estimation and Gaussian white noise." Ann. Statist. 38 (1) 181 - 214, February 2010. https://doi.org/10.1214/09-AOS705

Information

Published: February 2010
First available in Project Euclid: 31 December 2009

zbMATH: 1181.62152
MathSciNet: MR2589320
Digital Object Identifier: 10.1214/09-AOS705

Subjects:
Primary: 62G07 , 62G20

Keywords: ‎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

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 1 • February 2010
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