Cramer-von Mises test for beta distribution

Description

Performs Cramer-von Mises goodness-of-fit test for the hypothesis that the sample comes from a beta distribution. The statistic accumulates squared differences between the empirical distribution function and the theoretical beta cumulative distribution function.

Hypothesis of Beta Distribution The null hypothesis is that the data comes from a beta distribution with positive shape parameters alpha and beta on the interval \([0, 1]\).

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

Usage

from pysatl_criterion.statistics.goodness_of_fit import (
    CrammerVonMisesBetaGofStatistic,
)


test_statistic = CrammerVonMisesBetaGofStatistic(alpha=2, beta=5)
statistic_result = test_statistic.execute_statistic([0.08, 0.14, 0.22, 0.31, 0.38, 0.46, 0.57])
print(statistic_result)

Arguments

alpha - first positive shape parameter of the beta distribution. Default value is 1.

beta - second positive shape parameter of the beta 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 \(F_0\) is the reference beta cumulative distribution function.

Author(s)

Dmitry Deruzhinsky, Aleksei Tokarev, Vladimir Zakharov, 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 (
    CrammerVonMisesBetaGofStatistic,
)


test_statistic = CrammerVonMisesBetaGofStatistic(alpha=2, beta=5)
statistic_result = test_statistic.execute_statistic([0.08, 0.14, 0.22, 0.31, 0.38, 0.46, 0.57])
print(statistic_result)