On the Exact and Approximate Sampling Distribution of the Two Sample Kolmogorov-Smirnov Criterion D _<i>mn</i> , <i>m</i> ≤ <i>n</i>

Abstract

Abstract The size of errors of the Smirnov approximation to the exact null distribution of the two sample Kolmogorov-Smirnov criterion Dmn, m≤n is investigated. It is shown that a sample size n of 100 is necessary before the overestimating effect of the Smirnov distribution becomes negligible. Several improvements over the Smirnov approximation are considered. With continuity correction of order of , the Smirnov approximation accomplishes for n = 50 about what it does for n = 100 without continuity correction. For n ≥ 25, m/n≥.10, a competitor superior to the Smirnov approximation and having an approximately normal distribution is shown to exist through a logarithmic transformation of Dmn. For n≥50, m/n≤.10, the Kolmogorov one sample results are shown to give a better approximation than the competitor. Numerical results are included to show that the normal approximation is on the average as good as or somewhat better than the Smirnov approximation with continuity correction.