Cramer-von Mises test for Student distribution

Description

Performs Cramer-von Mises goodness-of-fit test for the hypothesis that the sample comes from a Student's t-distribution. The statistic accumulates squared differences between empirical plotting positions and Student cumulative distribution function values.

Hypothesis of Student Distribution The null hypothesis is that the data comes from a Student's t-distribution with positive degrees of freedom df, location parameter loc, and positive scale parameter scale.

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

Usage

from pysatl_criterion.statistics.goodness_of_fit import (
    CramerVonMisesStudentGofStatistic,
)


test_statistic = CramerVonMisesStudentGofStatistic(df=5, loc=0, scale=1)
statistic_result = test_statistic.execute_statistic([-1.8, -0.9, -0.25, 0.0, 0.31, 0.95, 1.7])
print(statistic_result)

Arguments

df - positive degrees of freedom of the Student's t-distribution. Default value is 1.

loc - location parameter of the Student's t-distribution. Default value is 0.

scale - positive scale parameter of the Student's t-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 Student cumulative distribution function after standardization.

Author(s)

Dmitriy Rusanov, 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 (
    CramerVonMisesStudentGofStatistic,
)


test_statistic = CramerVonMisesStudentGofStatistic(df=5, loc=0, scale=1)
statistic_result = test_statistic.execute_statistic([-1.8, -0.9, -0.25, 0.0, 0.31, 0.95, 1.7])
print(statistic_result)