# Cramér–von Mises criterion

Cramér–von Mises criterion

In statistics the Cramér–von Mises criterion is a criterion used for judging the goodness of fit of a cumulative distribution function F * compared to a given empirical distribution function Fn, 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

$\omega^2 = \int_{-\infty}^{\infty} [F_n(x)-F^*(x)]^2\,\mathrm{d}F^*(x)$

In one-sample applications F * is the theoretical distribution and Fn 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)

Let $x_1,x_2,\cdots,x_n$ be the observed values, in increasing order. Then the statistic is[1]:1153[2]

$T = n \omega^2 = \frac{1}{12n} + \sum_{i=1}^n \left[ \frac{2i-1}{2n}-F(x_i) \right]^2.$

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

### Watson test

A modified version of the Cramér–von Mises test is the Watson test[3] which uses the statistic U2, where[2]

$U^2= T-n( \bar{F}-\tfrac{1}{2} )^2,$

where

$\bar{F}=\frac{1}{n} \sum F(x_i).$

## Cramér–von Mises test (two samples)

Let $x_1,x_2,\cdots,x_N$ and $y_1,y_2,\cdots,y_M$ be the observed values in the first and second sample respectively, in increasing order. Let $r_1,r_2,\cdots,r_N$ be the ranks of the x's in the combined sample, and let $s_1,s_2,\cdots,s_M$ be the ranks of the y's in the combined sample. Anderson[1]:1149 shows that

$T = N \omega^2 = \frac{U}{N M (N+M)}-\frac{4 M N - 1}{6(M+N)}$

where U is defined as

$U = N \sum_{i=1}^N (r_i-i)^2 + M \sum_{j=1}^M (s_j-j)^2$

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 x, y, and r sequences. So xi is unique, and its rank is i in the sorted list x1,...xN. If there are duplicates, and xi through xj 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 (i + j) / 2. In the above equations, in the expressions (rii)2 and (sjj)2, duplicates can modify all four variables ri, i, sj, and j.

## Notes

1. ^ a b c d Anderson (1962)
2. ^ a b Pearson & Hartley (1972) p 118
3. ^ Watson (1961)
4. ^ Ruymgaart (1980)

## References

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