Editing Kuiper's test (section)

==Example==

We could test the hypothesis that computers fail more during some times of the year than others. To test this, we would collect the dates on which the test set of computers had failed and build an [[empirical distribution function]]. The [[null hypothesis]] is that the failures are [[Uniform distribution (continuous)|uniformly distributed]]. Kuiper's statistic does not change if we change the beginning of the year and does not require that we bin failures into months or the like.<ref name=K1960/><ref name=W1>Watson, G.S. (1961) "Goodness-Of-Fit Tests on a Circle", ''[[Biometrika]]'', 48 (1/2), 109–114 {{JSTOR|2333135}}</ref> Another test statistic having this property is the Watson statistic,<ref name=W1/><ref>[[Egon Pearson|Pearson, E.S.]], Hartley, H.O. (1972) ''Biometrika Tables for Statisticians, Volume 2'', CUP. {{isbn|0-521-06937-8}} (Page 118)</ref> which is related to the [[Cramér–von Mises criterion|Cramér–von Mises test]].

However, if failures occur mostly on weekends, many uniform-distribution tests such as K-S and Kuiper would miss this, since weekends are spread throughout the year. This inability to distinguish distributions with a comb-like shape from [[continuous uniform distribution]]s is a key problem with all statistics based on a variant of the K-S test. Kuiper's test, applied to the event times modulo one week, is able to detect such a pattern. Using event times that have been modulated with the K-S test can result in different results depending on how the data is phased. In this example, the K-S test may detect the non-uniformity if the data is set to start the week on Saturday, but fail to detect the non-uniformity if the week starts on Wednesday.