scipy.stats.cramervonmises

scipy.stats.cramervonmises(rvs, cdf, args=())

Perform the one-sample Cramér-von 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 1-D 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

Cramér-von Mises statistic.

pvaluefloat

The p-value.

See also

kstest, cramervonmises_2samp

Notes

New in version 1.6.0.

The p-value relies on the approximation given by equation 1.8 in [2]. It is important to keep in mind that the p-value 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 p-value is not reliable.

References

1

Cramér-von Mises criterion, Wikipedia, https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93von_Mises_criterion

2

Csörgő, S. and Faraway, J. (1996). The Exact and Asymptotic Distribution of Cramér-von Mises Statistics. Journal of the Royal Statistical Society, pp. 221-234.

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.

>>> from scipy import stats
>>> rng = np.random.default_rng()
>>> x = stats.norm.rvs(size=500, random_state=rng)
>>> res = stats.cramervonmises(x, 'norm')
>>> res.statistic, res.pvalue
(0.49121480855028343, 0.04189256516661377)

The p-value 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 samples 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
(0.07400330012187435, 0.7274595666160468)

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 its cdf method as an argument.

>>> frozen_dist = stats.norm(loc=2)
>>> res = stats.cramervonmises(y, frozen_dist.cdf)
>>> res.statistic, res.pvalue
(0.07400330012187435, 0.7274595666160468)

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 p-value 0.04 is less than our chosen significance level.