Cramér von Mises test

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

Test description

To compare the frequency distribution of a sample with a given hypothetical probability distribution, the test variable is calculated from the ascending sorted sample values and the distribution function of the given probability distribution according to the formula ${\ displaystyle T}$ ${\ displaystyle x_ {1}, x_ {2}, \ ldots, x_ {n}}$ ${\ displaystyle F}$

{\ displaystyle T = n \ omega ^ {2} = {\ frac {1} {12n}} + \ sum _ {i = 1} ^ {n} \ left [{\ frac {2i-1} {2n} } -F (x_ {i}) \ right] ^ {2}}

.

The p-value results from the comparison of the test variable with corresponding table values . The null hypothesis of the test 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 smaller than a predefined level of significance (for example 0.05) is thus to be interpreted as a statistically significant deviation of the distribution of the sample values from the predefined probability distribution. ${\ displaystyle T}$

To compare the frequency distributions of two different samples, the test size is calculated according to the formulas ${\ displaystyle T}$

{\ displaystyle T = N \ omega ^ {2} = {\ frac {U} {NM (N + M)}} - {\ frac {4MN-1} {6 (M + N)}}}

With

{\ displaystyle U = N \ sum _ {i = 1} ^ {N} (r_ {i} -i) ^ {2} + M \ sum _ {j = 1} ^ {M} (s_ {j} - j) ^ {2}}

.

The values in the first and the values in the second sample as well as the ranks of the values of the first sample and the ranks of the values of the second sample are sorted in ascending order in a common ranking of both samples. ${\ displaystyle x_ {1}, x_ {2}, \ ldots, x_ {N}}$ ${\ displaystyle y_ {1}, y_ {2}, \ ldots, y_ {M}}$ ${\ displaystyle r_ {1}, r_ {2}, \ ldots, r_ {N}}$ ${\ displaystyle s_ {1}, s_ {2}, \ ldots, s_ {M}}$

The p-value is obtained in a similar way to the one-sample case by comparing the test variable with corresponding tables. The null hypothesis of the Cramér von Mises test in the two-sample case is the assumption that the frequency distributions of the two samples do not differ. A p-value smaller than a given significance level (for example 0.05) therefore means a statistically significant difference between the distributions of the values of the two samples. ${\ displaystyle T}$

Alternative procedures

The Kolmogorov-Smirnov test provides both for the one-sample case and for the two-sample case, an alternative to the Cramér-von Mises criterion is, the latter but-sample case Two particularly for the more selective applies . Another alternative to the Cramér von Mises test for the one-sample case is the Anderson-Darling test . For the special application as a normality test , the Shapiro-Wilk test , the Jarque-Bera test and the Lilliefors test can also be used as alternative methods.

literature

Theodore Wilbur Anderson: On the Distribution of the Two-Sample Cramer-von Mises Criterion. In: The Annals of Mathematical Statistics. 33 (3 )/1962. Institute of Mathematical Statistics, ISSN 0003-4851 , pp. 1148-1159
Cramér-von-Mises test. In: Zakkula Govindarajulu: Nonparametric Inference. World Scientific, Hackensack NJ 2007, ISBN 9-81-270034-X , pp. 187-189