Cramér-von Mises test

2020 Mathematics Subject Classification: Primary: 62G10 [MSN][ZBL]

A 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 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 [C] and R. von Mises [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 "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 [AD]).

References

[C]	H. Cramér, "Sannolikhetskalkylen och nåcgra av dess användningar" , Stockholm (1926)
[M]	R. von Mises, "Mathematical theory of probability and statistics" (1964) (Translated from German)
[S]	N.V. Smirnov, "On the $\omega^2$-distribution of von Mises" Mat. Sb. , 2 : 5 (1937) pp. 973–993 (In Russian) (French abstract)
[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)
[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 [C] is [C2].

References

[C2]	H. Cramér, "On the composition of elementary errors II" Skand. Aktuarietidskr. (1928) pp. 171–280