Cramer-von Mises test for uniformity

Description

Performs the Cramer-von Mises goodness-of-fit test for the hypothesis of uniformity on the interval \([a, b]\). The statistic accumulates squared differences between the empirical distribution function and the theoretical uniform cumulative distribution function.

Hypothesis of Uniformity The null hypothesis is that the sample comes from a uniform distribution on the interval \([a, b]\).

Test Statistic The observations are sorted, transformed by the reference uniform cumulative distribution function, and compared with expected uniform plotting positions.

Usage

from pysatl_criterion.statistics.goodness_of_fit import (
    CrammerVonMisesUniformGofStatistic,
)


test_statistic = CrammerVonMisesUniformGofStatistic(a=0, b=1)
statistic_result = test_statistic.execute_statistic([0.12, 0.25, 0.41, 0.53, 0.77, 0.91])
print(statistic_result)

Arguments

a - left boundary of the uniform distribution. Default value is 0.

b - right boundary of the uniform distribution. Default value is 1.

rvs - array-like sample data passed to execute_statistic.

Details

The Cramer-von Mises statistic is

\[ W_n^2 = \frac{1}{12n} + \sum_{i=1}^{n} \left(F_0(X_{(i)}) - \frac{2i - 1}{2n}\right)^2 \]

where \(X_{(i)}\) is the \(i\)-th order statistic and \(F_0\) is the reference uniform cumulative distribution function. Large values indicate stronger deviation from the uniform model.

Author(s)

Aleksandr Podmarev, Alexey Mironov

References

Cramer, H. (1928): On the composition of elementary errors. - Skandinavisk Aktuarietidskrift, vol. 11, pp. 141-180.

von Mises, R. (1931): Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und theoretischen Physik. - Leipzig: Deuticke.

Examples

from pysatl_criterion.statistics.goodness_of_fit import (
    CrammerVonMisesUniformGofStatistic,
)


test_statistic = CrammerVonMisesUniformGofStatistic(a=0, b=1)
statistic_result = test_statistic.execute_statistic([0.12, 0.25, 0.41, 0.53, 0.77, 0.91])
print(statistic_result)