Kuiper's $V_n$ statistic, a measure for comparing the difference of ideal distribution and empirical distribution, is of great significance in the goodness-of-fit test. However, Kuiper's formulae for computing the cumulative distribution function, false positive probability and the upper tail quantile of $V_n$can not be applied to the case of small sample capacity $n$ since the approximation error is $\mathcal{O}(n^{-1})$. In this work, our contributions lie in three perspectives: firstly the approximation error is reduced to $\mathcal{O}(n^{-(k+1)/2})$ where $k$ is the expansion order with the \textit{high order expansion} (HOE) for the exponent of differential operator; secondly, a novel high order formula with approximation error $\mathcal{O}(n^{-3})$ is obtained by massive calculations; thirdly, the fixed-point algorithms are designed for solving the Kuiper pair of critical values and upper tail quantiles based on the novel formula. The high order expansion method for Kuiper's $V_n$-statistic is applicable for various applications where there are more than $5$ samples of data. The principles, algorithms and code for the high order expansion method are attractive for the goodness-of-fit test.

翻译：Kuiper $V_n$统计量作为衡量理想分布与经验分布差异的指标，在拟合优度检验中具有重要价值。然而，Kuiper公式在计算累积分布函数、假阳性概率及$V_n$上尾分位数时，由于近似误差为$\mathcal{O}(n^{-1})$，无法应用于小样本容量$n$的情形。本文的贡献体现在三个方面：首先，通过微分算子指数的\textit{高阶展开}（HOE），将近似误差降至$\mathcal{O}(n^{-(k+1)/2})$，其中$k$为展开阶数；其次，通过大量计算得到近似误差为$\mathcal{O}(n^{-3})$的新型高阶公式；最后，基于该公式设计了求解Kuiper临界值对及上尾分位数的定点算法。Kuiper $V_n$统计量的高阶展开方法适用于数据样本数超过$5$的各类应用场景。该方法的原理、算法及代码对拟合优度检验具有重要参考价值。