Editing Kuiper's test (section)

==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>