Cramér–von Mises test

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An alternative to the Kolmogorov–Smirnov test for testing the hypothesis that a set of data come from a specified continuous distribution. The test was suggested independently by Cramér in 1928 and von Mises in 1931. The test statistic W (sometimes written as W2) is formally defined by , where F0(x) is the distribution function specified by the null hypothesis, Fn(x) is the sample distribution function, and f0(x)=F0(x). In practice the statistic is calculated using , where , and x(j) is the jth ordered observation (x(1)x(2)≤…≤x(n)).

The test has been adapted for use with discrete random variables, for cases where parameters have to be estimated from the data, and for comparing two samples. A modification leads to the Anderson–Darling test.

Subjects: Probability and Statistics.

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