Cramér–von Mises criterion

The Cramér - von Mises test is a statistical test that can be examined whether the frequency distribution of data from a sample of a given hypothetical probability distribution is different ( one-sample case), or whether the frequency distributions of two different samples differ ( two-sample case). When comparing the distribution of a sample with the normal distribution, the process acts as a normality test. The test is named after Harald Cramér and Richard von Mises, who developed it 1928-1930 and published. The generalization for the two -sample case was described in 1962 by Theodore Wilbur Anderson.

Test Description

To compare the frequency distribution of a sample with a given hypothetical probability distribution of the test statistic from the ascending sorted sample values ​​and the distribution function of the given probability distribution calculated by the formula

By comparing the test statistic with the corresponding table values ​​yields the p-value. The null hypothesis of the tests in the one-sample case is the assumption that the distribution of the sample data does not differ from the given probability distribution. A p-value less than 0.05 is thus to be interpreted as statistically significant deviation in the distribution of the sample values ​​of the predetermined probability distribution.

To compare the frequency distributions of two different samples, the test statistic calculated using the formulas

With

Here are sorted in ascending order, respectively, the values ​​in the first and the values ​​in the second sample and the ranks of the values ​​in the first sample and the ranks of the values ​​of the second sample in a joint ranking in both samples.

The p- value is analogous to the one-sample case by comparing the test statistic with appropriate tables. The null hypothesis of the Cramér - von Mises tests in the two -sample case, the assumption that the frequency distributions are not different in both samples. Therefore, a p- value less than 0.05 indicates a statistically significant difference between the distributions of the values ​​of the two samples.

Alternative methods

The Kolmogorov -Smirnov test provides both for the one-sample case and for the two -sample case, an alternative to the Cramér -von- Mises test represents, the latter but especially true for the two-sample case as a test more. Another alternative to the Cramér - von Mises test for the one-sample case is the Anderson - Darling test. For the particular application as a normality test and the Shapiro - Wilk test, the Jarque - Bera test and the Lilliefors test can be used as an alternative method among others.