Editing Kuiper's test

{{Short description|Statistical test}}
'''Kuiper's test''' is used in [[statistics]] to [[statistical hypothesis test|test]] that whether a data sample come from a given [[cumulative distribution function|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]].<ref name=K1960>{{cite journal
 | last= Kuiper | first=N. H. |author-link=Nicolaas Kuiper
 | year = 1960
 | title = Tests concerning random points on a circle
 | journal = Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A
 | volume = 63
 | pages = 38–47
}}</ref>

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''<sup>+</sup> and ''D''<sup>&minus;</sup> represent the absolute sizes of the most positive and most negative differences between the two [[cumulative distribution function]]s that are being compared. The trick with Kuiper's test is to use the quantity ''D''<sup>+</sup>&nbsp;+&nbsp;''D''<sup>&minus;</sup> 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 [[seasonality|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 distribution|circular probability distributions]].

==One-sample Kuiper test==
[[File:KuiperTestVisualization 2Sample.png|thumb|300px|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.]]

The one-sample test statistic, <math>V_n</math>, 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 ''F''<sub>''n''</sub> the [[empirical distribution function]] for ''n'' [[Independent and identically distributed random variables|independent and identically distributed]] (i.i.d.) observations ''X<sub>i</sub>'', which is defined as 

:<math>
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}),
</math>
:where <math>1_{(-\infty,x]}(X_i)</math> is the [[indicator function]], equal to 1 if <math>X_i \le x</math> and equal to 0 otherwise.
Then the one-sided [[Kolmogorov–Smirnov test|Kolmogorov–Smirnov]] [[statistic]] for the given [[cumulative distribution function]] ''F''(''x'') is
:<math>D^+_n =  \sup_x [F_n(x)-F(x)],</math>
:<math>D^-_n =  \sup_x [F(x)-F_n(x)],</math>
where <math>\sup</math> is the [[Infimum and supremum|supremum function]]. And finally the one-sample Kuiper test is defined as,
:<math>V_n=D^+_n + D^-_n ,</math>
or equivalently
:<math>V_n=\sup_x [F_n(x)-F(x)] - \inf_x [F_n(x)-F(x)] ,</math>
where <math>\inf</math> is the [[Infimum and supremum|infimum function]].

Tables for the critical points of the test statistic <math>V_n</math> are available,<ref>[[Egon Pearson|Pearson, E.S.]], Hartley, H.O. (1972) ''Biometrika Tables for Statisticians, Volume 2'', CUP. {{isbn|0-521-06937-8}} (Table 54)</ref> and these include certain cases where the distribution being tested is not fully known, so that parameters of the family of distributions are [[estimation theory|estimated]].

The [[asymptotic distribution]] of the statistic <math>\sqrt{n}V_n</math> is given by,<ref name=K1960>{{cite journal
 | last= Kuiper | first=N. H. |author-link=Nicolaas Kuiper
 | year = 1960
 | title = Tests concerning random points on a circle
 | journal = Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A
 | volume = 63
 | pages = 38–47
}}</ref>

:<math>
\begin{align}
\operatorname{Pr}(\sqrt{n}V_n\leq x)=&1-2\sum_{k=1}^\infty (-1)^{k-1} (4k^2x^2-1)e^{-2k^2 x^2} \\
&+\frac{8}{3\sqrt{n}}x\sum_{k=1}^\infty k^2(4k^2x^2-3)e^{-2k^2 x^2}+\omicron(\frac{1}{n}).
\end{align}
</math>

For <math>x>\frac{6}{5}</math>, a reasonable approximation is obtained from the first term of the series as follows 

:<math>1 - 2(4x^2-1)e^{-2x^2}+\frac{8x}{3\sqrt{n}}(4x^2-3)e^{-2x^2}.</math>

==Two-sample Kuiper test==
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. In this case, the Kuiper statistic is

:<math>V_{n,m}=\sup_x [F_{1,n}(x)-F_{2,m}(x)]-\inf_x [F_{1,n}(x)-F_{2,m}(x)],</math>

where <math>F_{1,n}</math> and <math>F_{2,m}</math> are the [[empirical distribution function]]s of the first and the second sample respectively, <math>\sup</math> is the [[Infimum and supremum|supremum function]], and <math>\inf</math> is the [[Infimum and supremum|infimum function]].

==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.

==See also==
* [[Kolmogorov–Smirnov test]]

==References==
{{Reflist}}

[[Category:Statistical tests]]
[[Category:Nonparametric statistics]]
[[Category:Directional statistics]]
[[Category:1960 introductions]]