Fixed-Point Algorithm for Solving the Critical Value and Upper Tail Quantile of Kuiper's Statistics

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.