Yu Zhou

TL;DR: Thirteen dominant methods for estimating the Hurst exponent are put on for the purpose of decreasing the difficulty of implementing the estimation methods with computer programs, the mathematical principles are discussed briefly and the pseudo-codes of algorithms are presented with necessary details.

TL;DR: In this article , four key algorithms for computing CEI-1 are designed, verified, validated and tested, which can be utilized in R&D and be reused properly.

TL;DR: In this article , the authors propose a solution to solve the problem of the problem: this article ] of "uniformity" and "uncertainty" of the solution.

Abstract: Kuiper’s Vn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_n$$\end{document} statistic, a measure for comparing the difference of ideal distribution and empirical distribution, is of great significance in the goodness-of-fit test. However, Kuiper’s formulae for computing the cumulative distribution function, false positive probability, and the upper tail quantile of Vn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_n$$\end{document} cannot be applied to the case of small sample capacity n since the approximation error is On-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}\left( n^{-1}\right) $$\end{document}. In this work, our contributions lie in three perspectives: firstly the approximation error is reduced to On-(k+1)/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}\left( n^{-(k+1)/2}\right) $$\end{document} where k is the expansion order with the high order expansion for the exponent of the differential operator; secondly, a novel high order formula with approximation error On-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}\left( n^{-3}\right) $$\end{document} is obtained by massive calculations; thirdly, the fixed-point algorithms are designed for solving the Kuiper pair of critical values and upper tail quantiles based on the novel formula. The high order expansion method for Kuiper’s Vn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_n$$\end{document} statistic is applicable for various applications where there are more than five samples of data. The principles, algorithms, and code for the high order expansion method are attractive for the goodness-of-fit test.