scipy.stats.cramervonmises¶

scipy.stats.
cramervonmises
(rvs, cdf, args=())[source]¶ Perform the Cramérvon Mises test for goodness of fit.
This performs a test of the goodness of fit of a cumulative distribution function (cdf) \(F\) compared to the empirical distribution function \(F_n\) of observed random variates \(X_1, ..., X_n\) that are assumed to be independent and identically distributed ([1]). The null hypothesis is that the \(X_i\) have cumulative distribution \(F\).
 Parameters
 rvsarray_like
A 1D array of observed values of the random variables \(X_i\).
 cdfstr or callable
The cumulative distribution function \(F\) to test the observations against. If a string, it should be the name of a distribution in
scipy.stats
. If a callable, that callable is used to calculate the cdf:cdf(x, *args) > float
. argstuple, optional
Distribution parameters. These are assumed to be known; see Notes.
 Returns
 resobject with attributes
 statisticfloat
Cramérvon Mises statistic.
 pvaluefloat
The pvalue.
See also
Notes
New in version 1.6.0.
The pvalue relies on the approximation given by equation 1.8 in [2]. It is important to keep in mind that the pvalue is only accurate if one tests a simple hypothesis, i.e. the parameters of the reference distribution are known. If the parameters are estimated from the data (composite hypothesis), the computed pvalue is not reliable.
References
 1
https://en.wikipedia.org/wiki/Cramérvon_Mises_criterion
 2
Csorgo, S. and Faraway, J. (1996). The Exact and Asymptotic Distribution of Cramérvon Mises Statistics. Journal of the Royal Statistical Society, pp. 221234.
Examples
Suppose we wish to test whether data generated by
scipy.stats.norm.rvs
were, in fact, drawn from the standard normal distribution. We choose a significance level of alpha=0.05.>>> import numpy as np >>> from scipy import stats >>> np.random.seed(626) >>> x = stats.norm.rvs(size=500) >>> res = stats.cramervonmises(x, 'norm') >>> res.statistic, res.pvalue (0.06342154705518796, 0.792680516270629)
The pvalue 0.79 exceeds our chosen significance level, so we do not reject the null hypothesis that the observed sample is drawn from the standard normal distribution.
Now suppose we wish to check whether the same sampels shifted by 2.1 is consistent with being drawn from a normal distribution with a mean of 2.
>>> y = x + 2.1 >>> res = stats.cramervonmises(y, 'norm', args=(2,)) >>> res.statistic, res.pvalue (0.4798693195559657, 0.044782228803623814)
Here we have used the args keyword to specify the mean (
loc
) of the normal distribution to test the data against. This is equivalent to the following, in which we create a frozen normal distribution with mean 2.1, then pass itscdf
method as an argument.>>> frozen_dist = stats.norm(loc=2) >>> res = stats.cramervonmises(y, frozen_dist.cdf) >>> res.statistic, res.pvalue (0.4798693195559657, 0.044782228803623814)
In either case, we would reject the null hypothesis that the observed sample is drawn from a normal distribution with a mean of 2 (and default variance of 1) because the pvalue 0.04 is less than our chosen significance level.