Kuiper's statistic is a good measure for the difference of ideal distribution and empirical distribution in the goodness-of-fit test. However, it is a challenging problem to solve the critical value and upper tail quantile, or simply Kuiper pair, of Kuiper's statistics due to the difficulties of solving the nonlinear equation and reasonable approximation of infinite series. In this work, the contributions lie in three perspectives: firstly, the second order approximation for the infinite series of the cumulative distribution of the critical value is used to achieve higher precision; secondly, the principles and fixed-point algorithms for solving the Kuiper pair are presented with details; finally, finally, a mistake about the critical value $c^\alpha_n$ for $(\alpha, n)=(0.01,30)$ in Kuiper's distribution table has been labeled and corrected where $n$ is the sample capacity and $\alpha$ is the upper tail quantile. The algorithms are verified and validated by comparing with the table provided by Kuiper. The methods and algorithms proposed are enlightening and worth of introducing to the college students, computer programmers, engineers, experimental psychologists and so on.

翻译：Kuiper统计量是拟合优度检验中衡量理想分布与经验分布差异的良好度量。然而，由于求解非线性方程及对无穷级数进行合理近似的困难，求解Kuiper统计量的临界值与上尾分位数（简称Kuiper对）是一个具有挑战性的问题。本文的贡献体现在三个方面：首先，采用临界值累积分布无穷级数的二阶近似以实现更高精度；其次，详细阐述了求解Kuiper对的原理与不动点算法；最后，标记并修正了Kuiper分布表中关于$(\alpha, n)=(0.01,30)$的临界值$c^\alpha_n$的错误，其中$n$为样本容量，$\alpha$为上尾分位数。通过与Kuiper提供的表格进行比对，验证并确认了算法的有效性。本文提出的方法与算法具有启发性，值得向大学生、计算机程序员、工程师、实验心理学家等群体介绍。