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Statistical Theory for the Study of Nonstationary Time Series by Wavelet Methods

小波方法研究非平稳时间序列的统计理论

基本信息

批准号：
15530139
负责人：
TANAKA Katsuto
金额：
$ 1.47万
依托单位：
Hitotsubashi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

1.The wavelet method is unfamiliar to most statistician ; so I wrote an introductory, but intuitively appealing chapter about wavelets for statisticians, where it was emphasized that the wavelet method can deal with time and scale (frequency) at the same time. This is impossible if we rely on either the time-domain or frequency-domain method.2.We first considered classical time-domain and frequency domain methods for estimating nonstationary and long-memory time series models. Then these methods were compared with the wavelet method. It was found that the former behave better when time series is stationary, but superiority is reversed when time series is nonstationary. The difference becomes larger as the degree of nonstationarity increases.3.The long-memory model contaminated by noise, which is often referred to as the long-memory signal plus noise model, was used to test if there really exists noise, by using the wavelet method The test statistic was derived on the basis of the existing principle, which gave us a very simple form. The wavelet-based test in the present case is useful for complementing the existing method.4.We took up three representative models for testing the random walk hypothesis, and devised wavelet-based tests. One is the AR(1) model ; another the ARFIMA model, and the other the state-space model. The distributions of the test statistics, however, still remain to be derived. Here we approximated them by simulations.5.We discussed the distribution of quadratic functionals of the fractional Brownian motion, which appears as a weak limit when we deal with a long-memory process. This is also an unsolved problem, but we gave an intuitive idea about the shape of the distribution. We also gave a conjecture about the moment of the distribution, which is a very neat result.

1.大多数统计学家都不熟悉小波方法，所以我为统计学家写了一个介绍性的，但直观上很吸引人的关于小波的章节，其中强调小波方法可以同时处理时间和尺度（频率）。这是不可能的，如果我们依赖于时域或频域的方法。2.我们首先考虑了经典的时域和频域方法估计非平稳和长记忆时间序列模型。并与小波方法进行了比较。研究发现，当时间序列是平稳的时，前者表现得更好，但当时间序列是非平稳的时，前者的优越性就相反了。随着非平稳程度的增加，这种差异越来越大。3.利用小波方法，将噪声污染的长记忆模型，即通常所说的长记忆信号加噪声模型，用于检验是否真的存在噪声。本文提出的小波检验方法是对已有方法的一种补充。4.选取了三种具有代表性的随机游走假设检验模型，设计了小波检验方法。一个是AR（1）模型，另一个是ARFIMA模型，另一个是状态空间模型。然而，检验统计量的分布仍有待推导。5.讨论了分数布朗运动的二次泛函的分布，它在处理长记忆过程时表现为一个弱极限。这也是一个未解决的问题，但我们给出了关于分布形状的直观想法。我们还给出了一个关于分布矩的猜想，这是一个非常简洁的结果。