The Annals of Statistics

Asymptotic equivalence of spectral density estimation and Gaussian white noise

Georgi K. Golubev, Michael Nussbaum, and Harrison H. Zhou

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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.

Article information

Source
Ann. Statist. Volume 38, Number 1 (2010), 181-214.

Dates
First available in Project Euclid: 31 December 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1262271613

Digital Object Identifier
doi:10.1214/09-AOS705

Mathematical Reviews number (MathSciNet)
MR2589320

Zentralblatt MATH identifier
1181.62152

Subjects
Primary: 62G07: Density estimation 62G20: Asymptotic properties

Keywords
Stationary Gaussian process spectral density Sobolev classes Le Cam distance asymptotic equivalence Whittle likelihood log-periodogram regression nonparametric Gaussian scale model signal in Gaussian white noise

Citation

Golubev, Georgi K.; Nussbaum, Michael; Zhou, Harrison H. Asymptotic equivalence of spectral density estimation and Gaussian white noise. Ann. Statist. 38 (2010), no. 1, 181--214. doi:10.1214/09-AOS705. https://projecteuclid.org/euclid.aos/1262271613.


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