ASYMPTOTIC EQUIVALENCE OF SPECTRAL DENSITY ESTIMATION AND GAUSSIAN WHITE NOISE

作者：Golubev Georgi K^*; Nus**aum Michael; Zhou Harrison H

来源：Annals of Statistics, 2010, 38(1): 181-214.

DOI：10.1214/09-AOS705

• 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

We consider the statistical experiment given by a sample y(l),...,y(n) 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 f(omega(i)), where omega(i) is a uniform grid of points in (-pi, pi) (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.

• 出版日期2010-2