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 F ∗ {displaystyle F^{*}} compared to a given empirical distribution function F n {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 In statistics the Cramér–von Mises criterion is a criterion used for judging the goodness of fit of a cumulative distribution function F ∗ {displaystyle F^{*}} compared to a given empirical distribution function F n {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 In one-sample applications F ∗ {displaystyle F^{*}} is the theoretical distribution and F n {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. The generalization to two samples is due to Anderson. The Cramér–von Mises test is an alternative to the Kolmogorov–Smirnov test. Let x 1 , x 2 , ⋯ , x n {displaystyle x_{1},x_{2},cdots ,x_{n}} be the observed values, in increasing order. Then the statistic is:1153 If this value is larger than the tabulated value, then the hypothesis that the data came from the distribution F {displaystyle F} can be rejected. A modified version of the Cramér–von Mises test is the Watson test which uses the statistic U2, where