# 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 ${\displaystyle F^{*}}$ compared to a given empirical distribution function ${\displaystyle F_{n}}$, 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

${\displaystyle \omega ^{2}=\int _{-\infty }^{\infty }[F_{n}(x)-F^{*}(x)]^{2}\,\mathrm {d} F^{*}(x)}$

In one-sample applications ${\displaystyle F^{*}}$ is the theoretical distribution and ${\displaystyle F_{n}}$ 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.[1][2] The generalization to two samples is due to Anderson.[3]

The Cramér–von Mises test is an alternative to the Kolmogorov–Smirnov test (1933).[4]

## Cramér–von Mises test (one sample)

Let ${\displaystyle x_{1},x_{2},\cdots ,x_{n}}$  be the observed values, in increasing order. Then the statistic is[3]:1153[5]

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

If this value is larger than the tabulated value, then the hypothesis that the data came from the distribution ${\displaystyle F}$  can be rejected.

### Watson test

A modified version of the Cramér–von Mises test is the Watson test[6] which uses the statistic U2, where[5]

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

where

${\displaystyle {\bar {F}}={\frac {1}{n}}\sum _{i=1}^{n}F(x_{i}).}$

## Cramér–von Mises test (two samples)

Let ${\displaystyle x_{1},x_{2},\cdots ,x_{N}}$  and ${\displaystyle y_{1},y_{2},\cdots ,y_{M}}$  be the observed values in the first and second sample respectively, in increasing order. Let ${\displaystyle r_{1},r_{2},\cdots ,r_{N}}$  be the ranks of the x's in the combined sample, and let ${\displaystyle s_{1},s_{2},\cdots ,s_{M}}$  be the ranks of the y's in the combined sample. Anderson[3]:1149 shows that

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

where U is defined as

${\displaystyle U=N\sum _{i=1}^{N}(r_{i}-i)^{2}+M\sum _{j=1}^{M}(s_{j}-j)^{2}}$

If the value of T is larger than the tabulated values,[3]: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 ${\displaystyle x}$ , ${\displaystyle y}$ , and ${\displaystyle r}$  sequences. So ${\displaystyle x_{i}}$  is unique, and its rank is ${\displaystyle i}$  in the sorted list ${\displaystyle x_{1},...x_{N}}$ . If there are duplicates, and ${\displaystyle x_{i}}$  through ${\displaystyle x_{j}}$  are a run of identical values in the sorted list, then one common approach is the midrank[7] method: assign each duplicate a "rank" of ${\displaystyle (i+j)/2}$ . In the above equations, in the expressions ${\displaystyle (r_{i}-i)^{2}}$  and ${\displaystyle (s_{j}-j)^{2}}$ , duplicates can modify all four variables ${\displaystyle r_{i}}$ , ${\displaystyle i}$ , ${\displaystyle s_{j}}$ , and ${\displaystyle j}$ .

## References

1. ^ Cramér, H. (1928). "On the Composition of Elementary Errors". Scandinavian Actuarial Journal. 1928 (1): 13–74. doi:10.1080/03461238.1928.10416862.
2. ^ von Mises, R. E. (1928). Wahrscheinlichkeit, Statistik und Wahrheit. Julius Springer.
3. ^ a b c d Anderson, T. W. (1962). "On the Distribution of the Two-Sample Cramer–von Mises Criterion" (PDF). Annals of Mathematical Statistics. Institute of Mathematical Statistics. 33 (3): 1148–1159. doi:10.1214/aoms/1177704477. ISSN 0003-4851. Retrieved June 12, 2009.
4. ^ A.N. Kolmogorov, "Sulla determinizione empirica di una legge di distribuzione" Giorn. Ist. Ital. Attuari , 4 (1933) pp. 83–91
5. ^ a b Pearson, E.S., Hartley, H.O. (1972) Biometrika Tables for Statisticians, Volume 2, CUP. ISBN 0-521-06937-8 (page 118 and Table 54)
6. ^ Watson, G.S. (1961) "Goodness-Of-Fit Tests on a Circle", Biometrika, 48 (1/2), 109-114 JSTOR 2333135
7. ^ 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.
• M. A. Stephens (1986). "Tests Based on EDF Statistics". In D'Agostino, R.B.; Stephens, M.A. (eds.). Goodness-of-Fit Techniques. New York: Marcel Dekker. ISBN 0-8247-7487-6.