cramervonmises_2samp — SciPy v1.15.3 Manual (original) (raw)

scipy.stats.

scipy.stats.cramervonmises_2samp(x, y, method='auto', *, axis=0, nan_policy='propagate', keepdims=False)[source]#

Perform the two-sample Cramér-von Mises test for goodness of fit.

This is the two-sample version of the Cramér-von Mises test ([1]): for two independent samples \(X_1, ..., X_n\) and\(Y_1, ..., Y_m\), the null hypothesis is that the samples come from the same (unspecified) continuous distribution.

Parameters:

xarray_like

A 1-D array of observed values of the random variables \(X_i\). Must contain at least two observations.

yarray_like

A 1-D array of observed values of the random variables \(Y_i\). Must contain at least two observations.

method{‘auto’, ‘asymptotic’, ‘exact’}, optional

The method used to compute the p-value, see Notes for details. The default is ‘auto’.

axisint or None, default: 0

If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If None, the input will be raveled before computing the statistic.

nan_policy{‘propagate’, ‘omit’, ‘raise’}

Defines how to handle input NaNs.

keepdimsbool, default: False

If this is set to True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the input array.

Returns:

resobject with attributes

statisticfloat

Cramér-von Mises statistic.

pvaluefloat

The p-value.

Notes

Added in version 1.7.0.

The statistic is computed according to equation 9 in [2]. The calculation of the p-value depends on the keyword method:

If method='auto', the exact approach is used if both samples contain equal to or less than 20 observations, otherwise the asymptotic distribution is used.

If the underlying distribution is not continuous, the p-value is likely to be conservative (Section 6.2 in [3]). When ranking the data to compute the test statistic, midranks are used if there are ties.

Beginning in SciPy 1.9, np.matrix inputs (not recommended for new code) are converted to np.ndarray before the calculation is performed. In this case, the output will be a scalar or np.ndarray of appropriate shape rather than a 2D np.matrix. Similarly, while masked elements of masked arrays are ignored, the output will be a scalar or np.ndarray rather than a masked array with mask=False.

References

[2] (1,2)

Anderson, T.W. (1962). On the distribution of the two-sample Cramer-von-Mises criterion. The Annals of Mathematical Statistics, pp. 1148-1159.

[3]

Conover, W.J., Practical Nonparametric Statistics, 1971.

Examples

Suppose we wish to test whether two samples generated byscipy.stats.norm.rvs have the same distribution. We choose a significance level of alpha=0.05.

import numpy as np from scipy import stats rng = np.random.default_rng() x = stats.norm.rvs(size=100, random_state=rng) y = stats.norm.rvs(size=70, random_state=rng) res = stats.cramervonmises_2samp(x, y) res.statistic, res.pvalue (0.29376470588235293, 0.1412873014573014)

The p-value exceeds our chosen significance level, so we do not reject the null hypothesis that the observed samples are drawn from the same distribution.

For small sample sizes, one can compute the exact p-values:

x = stats.norm.rvs(size=7, random_state=rng) y = stats.t.rvs(df=2, size=6, random_state=rng) res = stats.cramervonmises_2samp(x, y, method='exact') res.statistic, res.pvalue (0.197802197802198, 0.31643356643356646)

The p-value based on the asymptotic distribution is a good approximation even though the sample size is small.

res = stats.cramervonmises_2samp(x, y, method='asymptotic') res.statistic, res.pvalue (0.197802197802198, 0.2966041181527128)

Independent of the method, one would not reject the null hypothesis at the chosen significance level in this example.