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

Cramér–von Mises test (one sample)

Let be the observed values, in increasing order. Then the statistic is
If this value is larger than the tabulated value, then the hypothesis that the data came from the distribution can be rejected.

Watson test

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

Cramér–von Mises test (two samples)

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 shows that
where U is defined as
If the value of T is larger than the tabulated values, the hypothesis that the two samples come from the same distribution can be rejected. .
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 method: assign each duplicate a "rank" of. In the above equations, in the expressions and, duplicates can modify all four variables, , , and.