Editing Kolmogorov–Smirnov test (section)

==The Kolmogorov–Smirnov statistic in more than one dimension==
A distribution-free multivariate Kolmogorov–Smirnov goodness of fit test has been proposed by [[Ana Justel|Justel]], Peña and Zamar (1997).<ref>{{cite journal |last1=Justel |first1=A.|author1-link=Ana Justel |last2=Peña |first2=D. |last3=Zamar |first3=R. |year=1997 |title=A multivariate Kolmogorov–Smirnov test of goodness of fit |journal=Statistics & Probability Letters |volume=35 |issue=3 |pages=251–259 |doi=10.1016/S0167-7152(97)00020-5 |citeseerx=10.1.1.498.7631 }}</ref> The test uses a statistic which is built using Rosenblatt's transformation, and an algorithm is developed to compute it in the bivariate case.  An approximate test that can be easily computed in any dimension is also presented.

The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to [[multivariate statistics|multivariate data]]. This is not straightforward because the maximum difference between two joint [[cumulative distribution function]]s is not generally the same as the maximum difference of any of the complementary distribution functions. Thus the maximum difference will differ depending on which of <math>\Pr(X < x \land Y < y)</math> or <math>\Pr(X < x \land Y > y)</math> or any of the other two possible arrangements is used. One might require that the result of the test used should not depend on which choice is made.

One approach to generalizing the Kolmogorov–Smirnov statistic to higher dimensions which meets the above concern is to compare the cdfs of the two samples with all possible orderings, and take the largest of the set of resulting KS statistics.  In ''d'' dimensions, there are 2<sup>''d''</sup>&nbsp;−&nbsp;1 such orderings.  One such variation is due to Peacock<ref name="Peacock">{{cite journal  |author = Peacock J.A. |title = Two-dimensional goodness-of-fit testing in astronomy |journal = [[Monthly Notices of the Royal Astronomical Society]] |volume = 202 |issue = 3 |pages = 615–627  |year = 1983 |bibcode = 1983MNRAS.202..615P |doi=10.1093/mnras/202.3.615|doi-access = free }}</ref> 
(see also Gosset<ref>{{cite journal  
|author = Gosset E.
|title =  A three-dimensional extended Kolmogorov–Smirnov test as a useful tool in astronomy}
|journal = Astronomy and Astrophysics
|volume = 188
|issue = 1
|pages = 258–264
|year = 1987
|bibcode = 1987A&A...188..258G
}}</ref> 
for a 3D version)
and another to Fasano and Franceschini<ref name="Fasano">{{cite journal |author=Fasano, G. |author2=Franceschini, A. |year=1987 |title= A multidimensional version of the Kolmogorov–Smirnov test |journal= Monthly Notices of the Royal Astronomical Society |issn=0035-8711 |volume= 225 |pages= 155–170 |bibcode=1987MNRAS.225..155F |doi=10.1093/mnras/225.1.155|doi-access= free }}</ref> (see Lopes et al. for a comparison and computational details).<ref name="Lopes">{{cite conference |author=Lopes, R.H.C. |author2=Reid, I. |author3=Hobson, P.R.  |title= The two-dimensional Kolmogorov–Smirnov test |conference= XI International Workshop on Advanced Computing and Analysis Techniques in Physics Research |date= 23–27 April 2007 |location= Amsterdam, the Netherlands |url= http://dspace.brunel.ac.uk/bitstream/2438/1166/1/acat2007.pdf }}</ref> Critical values for the test statistic can be obtained by simulations, but depend on the dependence structure in the joint distribution.