# 臺灣博碩士論文加值系統

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 在本篇論文中，我們探討自我相似性質過程的適合度檢定，考慮兩個著名的自我相似過程：分數布朗運動和分數差分ARMA過程。自我相似過程的赫斯(Hurst)參數由Jones和Shen (2004)所提出的嵌入分支過程方法估計得到。此自我相似性質的適合度檢定是基於皮爾(Pearson)卡方檢定的統計量。我們將檢定統計量的虛無假設分佈以近似尺度調整的卡方分佈修正傳統卡方分佈的第一型誤差偏差問題。而檢定統計量的尺度參數和自由度可以經由迴歸模型得到。模擬結果也顯示我們所提出方法可以有效的改進檢定統計量的有限樣本的顯著水準。最後也進行了高頻財務資料和人類心率資料的實證分析。
 In this paper, we focus on the goodness of fit test for self-similar property of two well-known processes: the fractional Brownian motion and the fractional autoregressive integrated moving average process. The Hurst parameter of the self-similar process is estimated by the embedding branching process method proposed by Jones and Shen (2004). The goodness of fit test for self-similarity is based on the Pearson chi-square test statistic. We approximate the null distribution of the test statistic by a scaled chi-square distribution to correct the size bias problem of the conventional chi-square distribution. The scale parameter and degrees of freedom of the test statistic are determined via regression method. Simulations are performed to show the finite sample size and power of the proposed test. Empirical applications are conducted for the high frequency financial data and human heart rate data.
 1 Introduction 12 Theoretical background 32.1 Self-similar processes . . . . . . . . . . . . . . . . . . . . 32.1.1 Stationary increments of self-similar process . . . . . . . 32.2 Processes with self-similar properties . . . . . . . . . . . . 62.2.1 Brownian Motion and Fractional Brownian Motion . . . . . . . 62.2.2 Gaussian Fractional ARIMA (FARIMA) . . . . . . . . . . . . . 72.3 Simulation of the fractional Brownian motion . . . . . . . . . 83 Estimation of the Hurst parameter 113.1 Embedded branching process (EBP) . . . . . . . . . . . . . . 114 Goodness of fit test for self-similarity 144.1 The method of goodness of fit test . . . . . . . . . . . . . . 144.2 A modification of the goodness of fit test . . . . . . . . . . 155 Simulation study 165.1 Modification of the program . . . . . . . . . . . . . . . . . 175.2 Standardization . . . . . . . . . . . . . . . . . . . . . . . 176 Application 196.1 High frequency financial data . . . . . . . . . . . . . . . . 196.2 Human heart rate data . . . . . . . . . . . . . . . . . . . . 197 Conclusion 20References 21Appendix 23
 Bates, S. and McLaughlin, S. (1996). An investigation of the implusive nature of Ethernet data using stable distributions. In Proceedings of the 12th UK Performance Engineering Workshop (Edited by J. Hillston and R. Pooley), 17-32.Bates, S. and McLaughlin, S. (1997). Testing the Gaussian assumption for self-similar teletraffic models. IEEE Signal Processing Workshop on higher-Order Statistics, 21-23.Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.Chiang, P. J. (2006). A Study on the Estimation of the Parameter and Goodness of Fit Test for the Self-similar processes. Master thesis, Department of Applied Mathmatics, National Sun Yat-sen University, Kaohsiung, Taiwan.Coeurjolly, J.F. (2000). Simulation and identification of the fractional brownian motion: a bibliographical and comparative study. Journal of Statistical Software, 5(7),1-53. URL http://www.jstatsoft.org/v05/i07.Davis, R.B. and Harte, D.S. (1987). Tests for Hurst effect, Biometrika, 74, 95-101.Embrechts, P. and Maejima, M. (2002). Selfsimilar Processes. Princeton Series in Applied Mathematica, Princeton University Press.Feder, J. (1988). Fractals, Plenum Press, New York.Guo, C. Y. (2004). Studies in the electrocardiogram monitoring indices. Master thesis, Department of Applied Mathmatics, National Sun Yat-sen University, Kaohsiung, Taiwan.Hurst, H. E. (1951). Long term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers, 116, 770-799.Jones, O.D. and Shen, Y. (2004). Estimating the Hurst index of a self-similar process via the crossing tree. IEEE Signal Processing Letters, 11(4), 416-419.Kolmogorov, A.N. (1941) Local structure of turbulence in fluid for very large Reynolds numbers. Transl. in Turbulence. S.K.Friedlander and L.Topper (eds.) (1961), Interscience Publishers, New York, 151-155.Leland, W.E., Taqqu, M.S., Willinger, W. and Wilson, D.V. (1994). On the self-similar nature of Ethernet tra±c (extended version). ACM Transactions on Networking, 2, 1-14.Mandelbrot, B.B. and Wallis, J.R. (1969a) Computer experiments with fractional Gaussian noises. Water Resources Res., 5, 1, 228-267.Mandelbrot, B.B. and Wallis, J.R. (1969b) Some long -run properities of geophysical records. Water Resources Res., 5, 321-340.Peng, C.-K., Havlin, S., Simons M, and Goldberger, A.L. (1995). Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos, 5, 82-87.Tsay, R. S. (2005). Analysis of financial time series, 2nd Edition. Wiley, Hoboken, New Jersey.Sakalauskien, G. (2003). The Hurst Phenomenon in Hydrology. Environmental Research, Engineering and Management, 3, 16-20.Taqqu, M.S., Teverovsky, V., and Willinger, W. (1995). Estimators for long-range dependence: an empirical study. Fractals, 3, 4, 785-788.Wood, A., Chan, G. (1994). Simulation of stationary Gaussian processes in [0,1]^d, Journal of computational and graphical statistics, 3, 4, 409-432.
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 1 自我相似過程之參數估計及適合度檢定之研究

 1 葉戰備：〈傳統義利觀析〉，《孔學與人生》，第28期，1994年5月。 2 張正明：〈縱橫商界五百年的山西商人〉，《歷史月刊》第160期，2001年5月。 3 周安邦：〈以文化符號學之觀點剖析《大紅燈籠高高掛》的主題思想與文化意涵〉，逢甲大學人文社會學院《逢甲人文社會學報》，第11期，2005年12月。

 1 最佳訂單執行問題的強化學習演算法 2 生物資訊、無線通訊及經濟模型的統計推論 3 電腦斷層掃描圖像的分割研究： 肺和肺結節 4 基於高頻交易資料的模擬系統 5 最小化自動倉儲支援製造現場供貨延遲時間模式建立 6 台灣證券交易所高頻交易資料分析 7 以科技學科教學知識觀點探究英語線上輔導教師之教學歷程 8 檢定統計量具有主要支配項時其檢定力的逼近方法

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