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scipy.stats.

# cramervonmises

scipy.stats.cramervonmises(rvs, cdf, args=(), *, axis=0, nan_policy='propagate', keepdims=False)

  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 \(F\).

  The test statistic \(T\) is defined as in [1], where \(\omega^2\) is the Cramér-von Mises criterion and \(x_i\) are the observed values.

  \[T = n\omega^2 = \frac{1}{12n} + \sum_{i=1}^n \left[ \frac{2i-1}{2n} - F(x_i) \right]^2\]
  Parameters:
    rvs array_like

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

    cdf str 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`.

    args tuple, optional

      Distribution parameters. These are assumed to be known; see Notes.

    axis int 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.

      - `propagate`: if a NaN is present in the axis slice (e.g. row) along which the statistic is computed, the corresponding entry of the output will be NaN.

      - `omit`: NaNs will be omitted when performing the calculation. If insufficient data remains in the axis slice along which the statistic is computed, the corresponding entry of the output will be NaN.

      - `raise`: if a NaN is present, a `ValueError` will be raised.

    keepdims bool, 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:
    res object with attributes
      statisticfloat

        Cramér-von Mises statistic \(T\).

      pvaluefloat

        The p-value.

  See also

  `kstest`, `cramervonmises_2samp`

  Notes

  Added 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.

  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`.

  Array API Standard Support

  `cramervonmises` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ⛔
  PyTorch | ✅ | ⛔
  JAX | ⚠️ no JIT | ⛔
  Dask | ⛔ | n/a

  `cramervonmises` also accepts MArrays backed by the backends indicated above; masked values will be treated as though they were not present.

  See Support for the array API standard for more information.

  References

  [1] (1,2)

  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`.

      >>> import numpy as np
      >>> 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
      (0.1072085112565724, 0.5508482238203407)

  The p-value 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
      (0.8364446265294695, 0.00596286797008283)

  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
      (0.8364446265294695, 0.00596286797008283)

  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 is less than our chosen significance level.