Kuiper's test is used in statistics to test whether a data sample comes from a given distribution (one-sample Kuiper test), or whether two data samples came from the same unknown distribution (two-sample Kuiper test). It is named after Dutch mathematician Nicolaas Kuiper. Kuiper's test is closely related to the better-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 absolute sizes of the most positive and most negative differences between the two cumulative distribution functions that are 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 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.
One-sample Kuiper test
The one-sample test statistic, V n {\displaystyle V_{n}} , for Kuiper's test is defined as follows. Let F be the continuous cumulative distribution function which is to be the null hypothesis. Denote by Fn the empirical distribution function for n independent and identically distributed (i.i.d.) observations Xi, which is defined as
F n ( x ) = number of (elements in the sample ≤ x ) n = 1 n ∑ i = 1 n 1 ( − ∞ , x ] ( X i ) , {\displaystyle F_{n}(x)={\frac {{\text{number of (elements in the sample}}\leq x)}{n}}={\frac {1}{n}}\sum _{i=1}^{n}1_{(-\infty ,x]}(X_{i}),}
where 1 ( − ∞ , x ] ( X i ) {\displaystyle 1_{(-\infty ,x]}(X_{i})} is the indicator function, equal to 1 if X i ≤ x {\displaystyle X_{i}\leq x} and equal to 0 otherwise. Then the one-sided Kolmogorov–Smirnov statistic for the given cumulative distribution function F(x) is
D n + = sup x [ F n ( x ) − F ( x ) ] , {\displaystyle D_{n}^{+}=\sup _{x}[F_{n}(x)-F(x)],}
D n − = sup x [ F ( x ) − F n ( x ) ] , {\displaystyle D_{n}^{-}=\sup _{x}[F(x)-F_{n}(x)],}
where sup {\displaystyle \sup } is the supremum function. And finally the one-sample Kuiper test is defined as,
V n = D n + + D n − , {\displaystyle V_{n}=D_{n}^{+}+D_{n}^{-},}
or equivalently
V n = sup x [ F n ( x ) − F ( x ) ] − inf x [ F n ( x ) − F ( x ) ] , {\displaystyle V_{n}=\sup _{x}[F_{n}(x)-F(x)]-\inf _{x}[F_{n}(x)-F(x)],}
where inf {\displaystyle \inf } is the infimum function. Tables for the critical points of the test statistic V n {\displaystyle V_{n}} are available, and these include certain cases where the distribution being tested is not fully known, so that parameters of the family of distributions are estimated. The asymptotic distribution of the statistic n V n {\displaystyle {\sqrt {n}}V_{n}} is given by,
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