# Cramér–von Mises criterion

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In statistics the **Cramér–von Mises criterion** is a criterion used for judging the goodness of fit of a cumulative distribution function compared to a given empirical distribution function , or for comparing two empirical distributions. It is also used as a part of other algorithms, such as minimum distance estimation. It is defined as

In one-sample applications is the theoretical distribution and is the empirically observed distribution. Alternatively the two distributions can both be empirically estimated ones; this is called the two-sample case.

The criterion is named after Harald Cramér and Richard Edler von Mises who first proposed it in 1928-1930. The generalization to two samples is due to Anderson.^{[1]}

The Cramér–von Mises test is an alternative to the Kolmogorov-Smirnov test.

## Cramér–von Mises test (one sample)Edit

Let be the observed values, in increasing order. Then the statistic is^{[1]}^{:1153}^{[2]}

If this value is larger than the tabulated value the hypothesis that the data come from the distribution can be rejected.

### Watson testEdit

A modified version of the Cramér–von Mises test is the Watson test^{[3]} which uses the statistic *U*^{2}, where^{[2]}

where

## Cramér–von Mises test (two samples)Edit

Let and be the observed values in the first and second sample respectively, in increasing order. Let be the ranks of the x's in the combined sample, and let be the ranks of the y's in the combined sample. Anderson^{[1]}^{:1149} shows that

where U is defined as

If the value of T is larger than the tabulated values,^{[1]}^{:1154–1159} the hypothesis that the two samples come from the same distribution can be rejected. (Some books^{[specify]}

give critical values for U, which is more convenient, as it avoids the need to compute T via the expression above. The conclusion will be the same).

The above assumes there are no duplicates in the , , and sequences. So is unique, and its rank is in the sorted list . If there are duplicates, and through are a run of identical values in the sorted list, then one common approach is the *midrank* ^{[4]} method: assign each duplicate a "rank" of . In the above equations, in the expressions and , duplicates can modify all four variables , , , and .

## See alsoEdit

## NotesEdit

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Anderson (1962) - ↑
^{2.0}^{2.1}Pearson & Hartley (1972) p 118 - ↑ Watson (1961)
- ↑ Ruymgaart (1980)

## ReferencesEdit

- Anderson, TW (1962). On the Distribution of the Two-Sample Cramer–von Mises Criterion.
*The Annals of Mathematical Statistics***33**(3): 1148–1159. - M. A. Stephens (1986). "Tests Based on EDF Statistics" D'Agostino, R.B. and Stephens, M.A.
*Goodness-of-Fit Techniques*, New York: Marcel Dekker. - Pearson, E.S., Hartley, H.O. (1972)
*Biometrika Tables for Statisticians, Volume 2*, CUP. ISBN 0521069378 (page 118 and Table 54) - Ruymgaart, F. H., (1980) "A unified approach to the asymptotic distribution theory of certain midrank statistics". In:
*Statistique non Parametrique Asymptotique*, 1±18, J. P. Raoult (Ed.), Lecture Notes on Mathematics, No. 821, Springer, Berlin. - Watson, G.S. (1961) "Goodness-Of-Fit Tests on a Circle",
*Biometrika*, 48 (1/2), 109-114 Template:Jstor

## Further reading Edit

- Xiao, Y., A. Gordon, A. Yakovlev (January 2007). A C++ Program for the Cramér–von Mises Two-Sample Test.
*Journal of Statistical Software***17**(8).

## External linksEdit

- C-vM Two Sample Test (Documentation for performing the test using R
- Table of Critical values for 1 sample CvM test

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