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

TL;DR

This paper develops fixed-point algorithms and second-order approximations to accurately compute Kuiper's critical values and upper tail quantiles, correcting previous errors and enhancing precision for goodness-of-fit testing.

Contribution

It introduces new fixed-point algorithms and second-order approximations for Kuiper's distribution, correcting prior errors and improving computational accuracy.

Findings

Higher precision in critical value computation achieved

Algorithms validated against Kuiper's distribution table

Correction of previous critical value mistake

Abstract

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.