Kuiper's test

As with the K-S test, the discrepancy statistics D+ and D− represent the absolute sizes of the most positive and most negative differences between the two cumulative distribution functions that are being compared.

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.

This invariance under cyclic transformations makes Kuiper's test invaluable when testing for cyclic variations by time of year or day of the week or time of day, and more generally for testing the fit of, and differences between, circular probability distributions.

Let F be the continuous cumulative distribution function which is to be the null hypothesis.

observations Xi, which is defined as Then the one-sided Kolmogorov–Smirnov statistic for the given cumulative distribution function F(x) is where

And finally the one-sample Kuiper test is defined as, or equivalently where

, a reasonable approximation is obtained from the first term of the series as follows The Kuiper test may also be used to test whether a pair of random samples, either on the real line or the circle coming from a common but unknown distribution.

We could test the hypothesis that computers fail more during some times of the year than others.

This inability to distinguish distributions with a comb-like shape from continuous uniform distributions 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.

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.

Illustration of the two-sample Kuiper Test statistic. Red and blue lines each correspond to an empirical distribution function, and the black arrows show the points distances which sum to the Kuiper Statistic.