ASYMPTOTIC EQUIVALENCE OF SPECTRAL DENSITY ESTIMATION AND GAUSSIAN WHITE NOISE

• 成果类型: Article
• 署名作者: Golubev, Georgi K.; Nussbaum, Michael; Zhou, Harrison H.
• 署名单位: Aix-Marseille Universite; Cornell University; Yale University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/09-AOS705
• 发表日期: 2010
• 页码: 181-214
• 关键词: time-series analysis; Nonparametric Regression; additional observations

摘要

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.