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 ${\displaystyle F^{*}}$ 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]

Let ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ 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.

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

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

Let ${\displaystyle x_{1},x_{2},\ldots ,x_{N}}$ and ${\displaystyle y_{1},y_{2},\ldots ,y_{M}}$ be the observed values in the first and second sample respectively, in increasing order. Let ${\displaystyle r_{1},r_{2},\ldots ,r_{N}}$ be the ranks of the xs in the combined sample, and let ${\displaystyle s_{1},s_{2},\ldots ,s_{M}}$ be the ranks of the ys 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 ${\displaystyle x_{1},\ldots ,x_{N}}$. 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 ${\displaystyle j}$.

• 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.

• Xiao, Y.; A. Gordon; A. Yakovlev (January 2007). "A C++ Program for the Cramér–von Mises Two-Sample Test" (PDF). Journal of Statistical Software. 17 (8). doi:10.18637/jss.v017.i08. ISSN 1548-7660. OCLC 42456366. S2CID 54098783. Retrieved June 12, 2009.