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.
[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).
be the observed values, in increasing order.
Then the statistic is[3]: 1153 [5] If this value is larger than the tabulated value, then the hypothesis that the data came from the distribution
can be rejected.
A modified version of the Cramér–von Mises test is the Watson test[6] which uses the statistic U2, where[5] where Let
be the observed values in the first and second sample respectively, in increasing order.
be the ranks of the xs in the combined sample, and let
be the ranks of the ys in the combined sample.
Anderson[3]: 1149 shows that where U is defined as 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
sequences.
is unique, and its rank is
in the sorted list
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
In the above equations, in the expressions
, duplicates can modify all four variables