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 1
2 Theoretical background 3
2.1 Self-similar processes . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Stationary increments of self-similar process . . . . . . . 3
2.2 Processes with self-similar properties . . . . . . . . . . . . 6
2.2.1 Brownian Motion and Fractional Brownian Motion . . . . . . . 6
2.2.2 Gaussian Fractional ARIMA (FARIMA) . . . . . . . . . . . . . 7
2.3 Simulation of the fractional Brownian motion . . . . . . . . . 8
3 Estimation of the Hurst parameter 11
3.1 Embedded branching process (EBP) . . . . . . . . . . . . . . 11
4 Goodness of fit test for self-similarity 14
4.1 The method of goodness of fit test . . . . . . . . . . . . . . 14
4.2 A modification of the goodness of fit test . . . . . . . . . . 15
5 Simulation study 16
5.1 Modification of the program . . . . . . . . . . . . . . . . . 17
5.2 Standardization . . . . . . . . . . . . . . . . . . . . . . . 17
6 Application 19
6.1 High frequency financial data . . . . . . . . . . . . . . . . 19
6.2 Human heart rate data . . . . . . . . . . . . . . . . . . . . 19
7 Conclusion 20
References 21
Appendix 23

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