View source for Cramér-von Mises test

{{MSC|62G10}}

[[Category:Nonparametric inference]]

A [[Non-parametric test|non-parametric test]] for testing a hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c0270101.png" /> which states that independent and identically-distributed random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c0270102.png" /> have a given continuous distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c0270103.png" />. The Cramér–von Mises test is based on a statistic of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c0270104.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c0270105.png" /> is the [[Empirical distribution|empirical distribution]] function constructed from the sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c0270106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c0270107.png" /> is a certain non-negative function defined on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c0270108.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c0270109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701011.png" /> are integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701012.png" />. Tests of this type, based on the  "square metric" , were first considered by H. Cramér [[#References|[1]]] and R. von Mises [[#References|[2]]]. N.V. Smirnov proposed putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701013.png" />, and showed that in that case, if the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701014.png" /> is valid and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701015.png" />, the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701016.png" /> has in the limit an [[Chi-squared test| "omega-squared"  distribution]], independent of the hypothetical distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701017.png" />. A statistical test for testing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701018.png" /> based on the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701019.png" />, is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701020.png" /> (Cramér–von Mises–Smirnov) test, and the numerical value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701022.png" /> is found using the following representation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701023.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701024.png" /> is the variational series based on the sample <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701025.png" />. According to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701026.png" /> test with significance level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701027.png" />, the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701028.png" /> is rejected whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701030.png" /> is the upper <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701031.png" />-quantile of the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701032.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701033.png" />. T.W. Anderson and D.A. Darling proposed a similarly constructed test, based on the statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701034.png" /> (see [[#References|[5]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cramér,   "Sannolikhetskalkylen och nåcgra av dess användningar" , Stockholm  (1926)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. von Mises,   "Mathematical theory of probability and statistics"  (1964)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.V. Smirnov,   "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701035.png" />-distribution of von Mises"  ''Mat. Sb.'' , '''2''' :  5  (1937)  pp. 973–993  (In Russian)  (French abstract)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.N. Bol'shev,   N.V. Smirnov,   "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka  (1983)  (In Russian)  (Processed by L.S. Bark and E.S. Kedrova)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  T.W. Anderson,   D.A. Darling,   "Asymptotic theory of certain  "goodness-of-fit"  criteria based on stochastic processes"  ''Ann. of Math. Stat.'' , '''23'''  (1952)  pp. 193–212</TD></TR></table>



====Comments====
Usually, the choice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701036.png" /> is simply called the Cramér–von Mises test in Western literature. However, Smirnov first proposed making this choice and rewrote the statistic in the distribution-free form above. The limit distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701037.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701038.png" /> whatever the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701039.png" />. (The term  "square metric"  refers to the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701040.png" />, not to some choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701041.png" />.) Cramér actually considered the test with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701042.png" /> replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701043.png" />, while von Mises used <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027010/c02701044.png" />.

An alternative to [[#References|[1]]] is [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Cramér,   "On the composition of elementary errors II"  ''Skand. Aktuarietidskr.''  (1928)  pp. 171–280</TD></TR></table>