Difference between revisions of "Cramér-von Mises test"
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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 | 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 | ||
Revision as of 08:34, 8 February 2012
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 [1] and R. von Mises [2]. 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 [5]).
References
[1] | H. Cramér, "Sannolikhetskalkylen och nåcgra av dess användningar" , Stockholm (1926) |
[2] | R. von Mises, "Mathematical theory of probability and statistics" (1964) (Translated from German) |
[3] | N.V. Smirnov, "On the ![]() |
[4] | 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) |
[5] | 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 [1] is [a1].
References
[a1] | H. Cramér, "On the composition of elementary errors II" Skand. Aktuarietidskr. (1928) pp. 171–280 |
Cramér-von Mises test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cram%C3%A9r-von_Mises_test&oldid=20895