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Kuiper's $V_n$ 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 $V_n$can not be applied to the case of small sample capacity $n$ since the approximation error is $\mathcal{O}(n^{-1})$. In this work, our contributions lie in three perspectives: firstly the approximation error is reduced to $\mathcal{O}(n^{-(k+1)/2})$ where $k$ is the expansion order with the \textit{high order expansion} (HOE) for the exponent of differential operator; secondly, a novel high order formula with approximation error $\mathcal{O}(n^{-3})$ 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 $V_n$-statistic is applicable for various applications where there are more than $5$ samples of data. The principles, algorithms and code for the high order expansion method are attractive for the goodness-of-fit test., Comment: arXiv admin note: text overlap with arXiv:2308.09463

The Hurst exponent is a significant indicator for characterizing the self-similarity and long-term memory properties of time sequences. It has wide applications in physics, technologies, engineering, mathematics, statistics, economics, psychology and so on. Currently, available methods for estimating the Hurst exponent of time sequences can be divided into different categories: time-domain methods and spectrum-domain methods based on the representation of time sequence, linear regression methods and Bayesian methods based on parameter estimation methods. Although various methods are discussed in literature, there are still some deficiencies: the descriptions of the estimation algorithms are just mathematics-oriented and the pseudo-codes are missing; the effectiveness and accuracy of the estimation algorithms are not clear; the classification of estimation methods is not considered and there is a lack of guidance for selecting the estimation methods. In this work, the emphasis is put on thirteen dominant methods for estimating the Hurst exponent. 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. It is expected that the survey could help the researchers to select, implement and apply the estimation algorithms of interest in practical situations in an easy way., Comment: 46 pages, 8 figures, 4 tables, 24 algorithms with pseudo-codes

Zernike radial polynomials (ZRP) play a significant role in application areas such as optics design, imaging systems, and image processing systems. Currently, there are two kinds of numerical schemes for computing the ZRP automatically with computer programs: one is based on the definition in which the factorial operations may lead to the overflow problem and the high order derivatives are troublesome, and the other is based on recursion which is either unstable or with high computational complexity. In this paper, our emphasis is focused on exploring the balanced binary tree (BBT) schemes for computing the ZRP: firstly an elegant formulae for computation is established; secondly the recursive and iterative algorithms based-on BBT are proposed; thirdly the computational complexity of the algorithms are analyzed rigorously; finally the performance of BBT schemes by testing the running time is verified and validated. Theoretical analysis shows that the computational complexity of balanced binary tree recursive algorithm (BBRTA) and iterative algorithm are exponential and quadratic respectively, which coincides with the running time test very well. Experiments show that the time consumption is about $1\sim 10$ microseconds with different computation platforms for the balanced binary tree iterative algorithm (BBTIA), which is stable and efficient for real-time applications. In the sense of STEM education, the connection of the BBT and ZRP exhibits the beauty and applications of discrete mathematical structure behind the engineering problem, which is worthy of introducing to the college students, computer programmers and optics engineers., Comment: 17 pages, 4 figures