Computing the Cumulative Distribution Function and Quantiles of the One-sided Kolmogorov-Smirnov Statistic

Abstract

The cumulative distribution and quantile functions for the one-sided one sample Kolmogorov-Smirnov probability distributions are used for goodness-of-fit testing. While the Smirnov-Birnbaum-Tingey formula for the CDF appears straight forward, its numerical evaluation generates intermediate results spanning many hundreds of orders of magnitude and at times requires very precise accurate representations. Computing the quantile function for any specific probability may require evaluating both the CDF and its derivative, both of which are computationally expensive. To work around avoid these issues, different algorithms can be used across different parts of the domain, and approximations can be used to reduce the computational requirements. We show here that straight forward implementation incurs accuracy loss for sample sizes of well under 1000. Further the approximations in use inside the open source SciPy python software often result in increased computation, not just reduced accuracy, and at times suffer catastrophic loss of accuracy for any sample size. Then we provide alternate algorithms which restore accuracy and efficiency across the whole domain.