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. The pioneering work by Kuiper just provided the key ideas and few numerical tables created from the upper tail probability $\alpha$ and sample capacity $n$, which limited its propagation and possible applications in various fields since there are infinite configurations for the parameters $\alpha$ and $n$. 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, an error in Kuiper's table of critical value is discovered and fixed. The algorithms are verified and validated by comparing with the table provided by Kuiper. The methods and algorithms proposed are enlightening and worthy 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逼近存在困难，求解Kuiper统计量的临界值与上尾分位数（简称Kuiper对）是一项具有挑战性的课题。Kuiper的前驱性工作仅提供了关键思想及基于上尾概率$\alpha$和样本容量$n$生成的少量数值表格，这限制了该方法在多个领域的推广与潜在应用，因为参数$\alpha$和$n$存在无穷多种配置。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获得更高精度；其次，详细阐述了求解Kuiper对的原理与定点算法；最后，发现并修正了Kuiper临界值表格中的一处错误。通过与Kuiper提供的表格进行对比，验证了算法的有效性。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推广。