How Much Information Does Dependence Between Wavelet Coefficients Contain?

• 成果类型: Article
• 署名作者: Jentsch, Carsten; Kirch, Claudia
• 署名单位: University of Mannheim; Otto von Guericke University
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2015.1093945
• 发表日期: 2016
• 页码: 1330-1345
• 关键词: nonstationary time-series; linear process bootstrap; memory parameter; frequency-domain; BLOCK BOOTSTRAP; range; decorrelation; statistics; estimator; inference

摘要

This article is motivated by several articles that propose statistical inference where the independence of wavelet coefficients for both short- as well as long-range dependent time series is assumed. We focus on the sample variance and investigate the influence of the dependence between wavelet coefficients and this statistic. To this end, we derive asymptotic distributional properties of the sample variance for a time series that is synthesized, ignoring some or all dependence between wavelet coefficients. We show that the second-order properties differ from the those of the true time series whose wavelet coefficients have the same marginal distribution except in the independent Gaussian case. This,holds true even if the dependency is correct within each level and only the dependence between levels is ignored. In the case of sample auto-covariances and sample autocorrelations at lag one,, we indicate that first-order properties are erroneous. In a second step, we investigate several nonparametric bootstrap, schemes in the wavelet domain, which take more and more dependence into account until finally the full dependency is mimicked. We obtain very similar results, where only a bootstrap mimicking the full covariance structure correctly can be valid asymptotically. A simulation study supports our theoretical findings for the wavelet domain bootstraps. For long-range-dependent time series with long-memory parameter d > 1/4, we show that some additional problems occur, which cannot be solved easily without using additional information for the bootstrap. Supplementary materials for this article are available online.