Cramér-von Mises test

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

A non-parametric test for testing a hypothesis which states that independent and identically-distributed random variables have a given continuous distribution function . The Cramér–von Mises test is based on a statistic of the type

where is the empirical distribution function constructed from the sample and is a certain non-negative function defined on the interval such that , and are integrable on . 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 , and showed that in that case, if the hypothesis is valid and , the statistic has in the limit an "omega-squared" distribution, independent of the hypothetical distribution function . A statistical test for testing based on the statistic , is called an (Cramér–von Mises–Smirnov) test, and the numerical value of is found using the following representation:

where is the variational series based on the sample . According to the test with significance level , the hypothesis is rejected whenever , where is the upper -quantile of the distribution of , i.e. . T.W. Anderson and D.A. Darling proposed a similarly constructed test, based on the statistic (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 -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 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 is independent of whatever the choice of . (The term "square metric" refers to the expression , not to some choice of .) Cramér actually considered the test with replaced by , while von Mises used .

An alternative to [C] is [C2].

References

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