2004 Fiscal Year Final Research Report Summary
Statistical Theory for the Study of Nonstationary Time Series by Wavelet Methods
| Project/Area Number |
15530139
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Economic statistics
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| Research Institution | Hitotsubashi University |
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Principal Investigator |
TANAKA Katsuto Hitotsubashi University, Graduate School of Economics, Professor, 大学院・経済学研究科, 教授 (40126595)
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| Project Period (FY) |
2003 – 2004
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| Keywords | wavelet / long-memory time series / ARFIMA model / limiting distribution / test statistic / random walk / fractional Brownian motion / nonstationary time series |
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| Research Abstract |
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.
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Research Products
(9 results)
