View source for Cramér-von Mises test

{{TEX|done}}

{{MSC|62G10}}

[[Category:Nonparametric inference]]

A [[Non-parametric test|non-parametric test]] for testing a hypothesis  $  H _{0} $
which states that independent and identically-distributed random variables  $  X _{1} \dots X _{n} $
have a given continuous distribution function  $  F (x) $. 
The Cramér–von Mises test is based on a statistic of the type

$$ 
\omega _ n^{2} [ \Psi (F (x))] \  = \ 
\int\limits _ {- \infty} ^ {+ \infty}
\left [ \sqrt n (F _{n} (x) -
F (x)) \right ]^{2} \Psi (F (x)) \  dF (x),
$$

where  $  F _{n} (x) $
is the [[Empirical distribution|empirical distribution]] function constructed from the sample  $  X _{1} \dots X _{n} $
and  $  \Psi (t) $
is a certain non-negative function defined on the interval  $  [0,\  1] $
such that  $  \Psi (t) $, 
$  t \Psi (t) $
and  $  t^{2} \Psi (t) $
are integrable on  $  [0,\  1] $. 
Tests of this type, based on the  "square metric" , were first considered by H. Cramér {{Cite|C}} and R. von Mises {{Cite|M}}. N.V. Smirnov proposed putting  $  \Psi (t) \equiv 1 $, 
and showed that in that case, if the hypothesis  $  H _{0} $
is valid and  $  n \rightarrow \infty $, 
the statistic  $  \omega^{2} = \omega _ n^{2} $
has in the limit an [[Chi-squared test| "omega-squared"  distribution]], independent of the hypothetical distribution function  $  F (x) $. 
A statistical test for testing  $  H _{0} $
based on the statistic  $  \omega _ n^{2} $, 
is called an  $  \omega^{2} $(
Cramér–von Mises–Smirnov) test, and the numerical value of  $  \omega _ n^{2} $
is found using the following representation:

$$ 
\omega _ n^{2} \  = \ 
{
\frac{1}{12n}
} +
\sum _ {j = 1} ^ n
\left [ F (X _{(j)} ) -

\frac{2j - 1}{2n}

\right ]^{2}  $$

where  $  X _{(1)} \leq  \dots \leq  X _{(n)} $
is the variational series based on the sample  $  X _{1} \dots X _{n} $. 
According to the  $  \omega^{2} $
test with significance level  $  \alpha $, 
the hypothesis  $  H _{0} $
is rejected whenever  $  \omega _ n^{2} \geq \omega _ \alpha^{2} $, 
where  $  \omega _ \alpha^{2} $
is the upper  $  \alpha $-
quantile of the distribution of  $  \omega^{2} $, 
i.e.  $  {\mathsf P} \{ \omega^{2} < \omega _ \alpha^{2} \} = 1 - \alpha $. 
T.W. Anderson and D.A. Darling proposed a similarly constructed test, based on the statistic  $  \omega _ n^{2} [(1 - F (x))/F(x)] $(
see {{Cite|AD}}).

====References====
{|
|valign="top"|{{Ref|C}}|| H. Cramér,   "Sannolikhetskalkylen och nåcgra av dess användningar" , Stockholm  (1926)
|-
|valign="top"|{{Ref|M}}|| R. von Mises,   "Mathematical theory of probability and statistics"  (1964)  (Translated from German)
|-
|valign="top"|{{Ref|S}}|| N.V. Smirnov,   "On the $\omega^2$-distribution of von Mises"  ''Mat. Sb.'' , '''2''' :  5  (1937)  pp. 973–993  (In Russian)  (French abstract)
|-
|valign="top"|{{Ref|BS}}|| 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)
|-
|valign="top"|{{Ref|AD}}|| 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
|}

====Comments====
Usually, the choice  $  \Psi (t) \equiv 1 $
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  $  \omega _ n^{2} $
is independent of  $  F $
whatever the choice of  $  \Psi $. 
(The term  "square metric"  refers to the expression  $  [ \sqrt n (F _{n} (x) - F (x))]^{2} $, 
not to some choice of  $  \Psi $.) 
Cramér actually considered the test with  $  \Psi (F (x)) \  dF (x) $
replaced by  $  dx $, 
while von Mises used  $  \lambda (x) \  dx $.

An alternative to {{Cite|C}} is {{Cite|C2}}.

====References====
{|
|valign="top"|{{Ref|C2}}|| H. Cramér,   "On the composition of elementary errors II"  ''Skand. Aktuarietidskr.''  (1928)  pp. 171–280
|}