Kolmogorov-Smirnov Two-Sample Tests

We have not previously discussed the use of criteria suggested by direct comparison of empirical (sample) cumulative distribution functions with one another or with hypothetical c.d.f.’s (“goodness of fit”). This important approach leads to a wide variety of procedures which stand apart from the procedures of earlier chapters in several respects. They are expressed in a different form. The relevant statistics are not approximately or asymptotically normally distributed. The theory of their asymptotic behavior is fascinating and raises different kinds of problems requiring different kinds of tools. The mathematical interest of these and other problems has played a larger role than statistical questions in motivating the extensive literature about them, although there is also some excellent work on statistically important questions.