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 that solving the critical value and upper tail quantile, or simply Kuiper pair, of Kuiper's statistics due to difficulties of solving the nonlinear equation and reasonable approximation of infinite series. The pioneering work by Kuiper and Stephens just provided the key ideas and few numerical tables created from the 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 two perspectives: firstly, the second order approximation for the infinite series of the cumulative distribution of the critical value is used to get higher precision; secondly, the principles and fixed-point algorithms for solving the Kuiper pair are presented with details. The algorithms are verified and validated by comparison 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统计量是检验理想分布与经验分布拟合优度的有效度量指标。然而，由于非线性方程求解的困难以及无穷级数合理逼近的挑战，求解Kuiper统计量的临界值与上尾分位数（简称Kuiper对）成为棘手问题。Kuiper和Stephens的开创性工作仅提供了关键思想及基于上尾概率α和样本容量n生成的少量数值表格，但因参数α与n存在无限种配置组合，这限制了该方法在各类领域的推广与应用。本文的贡献体现在两方面：其一，采用临界值累积分布无穷级数的二阶近似以提高精度；其二，详细阐述了求解Kuiper对的原理与固定点算法。通过与Kuiper提供的数表进行对比验证，算法的正确性和有效性得到证实。本文提出的方法与算法具有启发性，值得向高校学生、计算机程序员、工程师、实验心理学家等群体推广。