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 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. 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. 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.

Further reading

• Xiao, Y.; A. Gordon; A. Yakovlev (January 2007). "A C++ Program for the Cramér–von Mises Two-Sample Test" (PDF). Journal of Statistical Software. American Statistical Association. 17 (8). ISSN 1548-7660. OCLC 42456366. Retrieved June 12, 2009.