# Kuiper's test

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In statistics, **Kuiper's test** is closely related to the more well-known Kolmogorov-Smirnov test (or K-S test as it is often called). As with the K-S test, the discrepancy statistics *D*^{+} and *D*^{−} represent the maximum deviation above and below of the two cumulative distribution functions being compared. The trick with Kuiper's test is to use the quantity *D*^{+} + *D*^{−} as the test statistic. This small change makes Kuiper's test as sensitive in the tails as at the median and also makes it invariant under cyclic transformations of the independent variable. The Anderson-Darling test is another test that provides equal sensitivity at the tails as the median, but it does not provide the cyclic invariance.

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.