Editing Kolmogorov–Smirnov test (section)

===Discrete and mixed null distribution===

Under the assumption that <math>F</math> is non-decreasing and right-continuous, with countable (possibly infinite) number of jumps, the KS test statistic can be expressed as:

<math display="block">D_n= \sup_x |F_n(x)-F(x)| = \sup_{0 \leq t \leq 1} |F_n(F^{-1}(t)) - F(F^{-1}(t))|.</math>

From the right-continuity of <math>F</math>, it follows that <math>F(F^{-1}(t)) \geq t</math> and <math>F^{-1}(F(x)) \leq x </math> and hence, the distribution of <math>D_{n}</math> depends on the null distribution <math>F</math>, i.e., is no longer distribution-free as in the continuous case. Therefore, a fast and accurate method has been developed to compute the exact and asymptotic distribution of <math>D_{n}</math> when <math>F</math> is purely discrete or mixed,<ref name=DKT2019/> implemented in C++ and in the KSgeneral package <ref name=KSgeneral/> of the [[R (programming language)|R language]]. The functions <code>disc_ks_test()</code>, <code>mixed_ks_test()</code> and <code>cont_ks_test()</code> compute also the KS test statistic and p-values for purely discrete, mixed or continuous null distributions and arbitrary sample sizes. The KS test and its p-values for discrete null distributions and small sample sizes are also computed in <ref name=arnold-emerson>{{Cite journal |first1=Taylor B. |last1=Arnold |first2=John W. |last2=Emerson |year=2011 |title=Nonparametric Goodness-of-Fit Tests for Discrete Null Distributions |journal=The R Journal |volume=3 |issue=2 |pages=34\[Dash]39 |url=http://journal.r-project.org/archive/2011-2/RJournal_2011-2_Arnold+Emerson.pdf |doi=10.32614/rj-2011-016|doi-access=free }}</ref> as part of the dgof package of the R language. Major statistical packages among which [[SAS (software)|SAS]] <code>PROC NPAR1WAY</code>,<ref>{{cite web|url=https://support.sas.com/documentation/cdl/en/statug/68162/HTML/default/viewer.htm#statug_npar1way_toc.htm|title=SAS/STAT(R) 14.1 User's Guide|website=support.sas.com|access-date=14 April 2018}}</ref> [[Stata]] <code>ksmirnov</code><ref>{{cite web|url=https://www.stata.com/manuals15/rksmirnov.pdf|title=ksmirnov — Kolmogorov–Smirnov equality-of-distributions test|website=stata.com|access-date=14 April 2018}}</ref> implement the KS test under the assumption that <math>F(x)</math> is continuous, which is more conservative if the null distribution is actually not continuous (see <ref name=Noether63>{{Cite journal |vauthors=Noether GE |year=1963|title=Note on the Kolmogorov Statistic in the Discrete Case |journal=Metrika |volume=7 |issue=1 |pages=115–116|doi=10.1007/bf02613966|s2cid=120687545}}</ref> 
<ref name=Slakter65>{{Cite journal |vauthors=Slakter MJ |year=1965|title=A Comparison of the Pearson Chi-Square and Kolmogorov Goodness-of-Fit Tests with Respect to Validity |journal=Journal of the American Statistical Association |volume=60 |issue=311 |pages=854–858 |doi=10.2307/2283251|jstor=2283251}}</ref> 
<ref name=Walsh63>{{Cite journal |vauthors=Walsh JE  |year=1963 |title=Bounded Probability Properties of Kolmogorov–Smirnov and Similar Statistics for Discrete Data |journal=Annals of the Institute of Statistical Mathematics |volume=15 |issue=1 |pages=153–158|doi=10.1007/bf02865912|s2cid=122547015 }}</ref>).