BAYESIAN NONPARAMETRIC ESTIMATION OF THE SPECTRAL DENSITY OF A LONG OR INTERMEDIATE MEMORY GAUSSIAN PROCESS

• 成果类型: Article
• 署名作者: Rousseau, Judith; Chopin, Nicolas; Liseo, Brunero
• 署名单位: Institut Polytechnique de Paris; ENSAE Paris; Sapienza University Rome
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS955
• 发表日期: 2012
• 页码: 964-995
• 关键词: time-series; posterior distributions; asymptotic expansions; convergence-rates; range dependence; stationary; parameter; regression

摘要

A stationary Gaussian process is said to be long-range dependent (resp., anti-persistent) if its spectral density f(lambda) can be written as f(lambda) = vertical bar lambda vertical bar(-2d) g(vertical bar lambda vertical bar), where 0 < 1/2 (resp., -1/2 < 0), and g is continuous and positive. We propose a novel Bayesian nonparametric approach for the estimation of the spectral density of such processes. We prove posterior consistency for both d and g, under appropriate conditions on the prior distribution. We establish the rate of convergence for a general class of priors and apply our results to the family of fractionally exponential priors. Our approach is based on the true likelihood and does not resort to Whittle's approximation.