This invariance under cyclic transformations makes Kuiper's test invaluable when testing for variations by time of year or day of the week or time of day. One example would be to 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 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.
However, this test tends to miss the fact that failures occur only on weekends, since weekends are spread throughout the year. This inability to distinguish distributions with a comb-like shape from continuous distributions is a key problem with all statistics based on a variant of the K-S test.