Rational Arithmetic Mathematica Functions to Evaluate the One-sided One-sample K-S Cumulative Sample Distribution

Abstract

One of the most widely used goodness-of-fit tests is the Kolmogorov-Smirnov (KS) family of tests which have been implemented by many computer statistical software packages. To calculate a p value (evaluate the cumulative sampling distribution), these packages use various methods including recursion formulae, limiting distributions, and approximations of unknown accuracy developed over thirty years ago. Based on an extensive literature search for the one-sided one-sample K-S test, this paper identifies two direct formulae and five recursion formulae that can be used to calculate a p value and then develops two additional direct formulae and four iterative versions of the direct formulae for a total of thirteen formulae. To ensure accurate calculation by avoiding catastrophic cancelation and eliminating rounding error, each formula is implemented in rational arithmetic. Linear search is used to calculate the inverse of the cumulative sampling distribution (find the confidence interval bandwidth). Extensive tables of bandwidths are presented for sample sizes up to 2, 000. The results confirm the hypothesis that as the number of digits in the numerator and denominator integers of the rational number test statistic increases, the computation time also increases. In comparing the computational times of the thirteen formulae, the direct formulae are slightly faster than their iterative versions and much faster than all the recursion formulae. Computational times for the fastest formula are given for sample sizes up to fifty thousand.