A comparison of the discrete Kolmogorov-Smirnov statistic and the Euclidean distance

Goodness-of-fit tests gauge whether a given set of observations is consistent (up to expected random fluctuations) with arising as independent and identically distributed (i.i.d.) draws from a user-specified probability distribution known as the “model.” The standard gauges involve the discrepancy between the model and the empirical distribution of the observed draws. Some measures of discrepancy are cumulative; others are not. The most popular cumulative measure is the Kolmogorov-Smirnov statistic; when all probability distributions under consideration are discrete, a natural noncumulative measure is the Euclidean distance between the model and the empirical distributions. In the present paper, both mathematical analysis and its illustration via various data sets indicate that the Kolmogorov-Smirnov statistic tends to be more powerful than the Euclidean distance when there is a natural ordering for the values that the draws can take — that is, when the data is ordinal — whereas the Euclidean distance is more reliable and more easily understood than the Kolmogorov-Smirnov statistic when there is no natural ordering (or partial order) — that is, when the data is nominal.