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

First available in Project Euclid: 31 December 2009

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Zentralblatt MATH identifier

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

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


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.

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