scipy.stats._quantile:estimated_cdf

Notes

Given a sample x from an underlying distribution, estimated_cdf provides a nonparametric estimate of the cumulative distribution function.

By default, this is done by computing the "fractional index" p at which y would appear within z, a sorted copy of x:

p = 1 / (n - 1) * (j +  (     y - z[j])
                      / (z[j+1] - z[j]))

where the index j is that of the largest element of z that does not exceed y, and n is the number of elements in the sample. Note that if y is an element of z, then j is the index such that y = z[j], and the formula simplifies to the intuitive j / (n - 1). The full formula linearly interpolates between j / (n - 1) and (j + 1) / (n - 1).

This is a special case of the more general

p = (j + (y - z[j]) / (z[j+1] - z[j] + 1 - m) / n

where m may be defined according to several different conventions. The preferred convention may be selected using the method parameter:

=============================== =============== ===============
``method``                      number in H&F   ``m``
=============================== =============== ===============
``interpolated_inverted_cdf``   4               ``0``
``hazen``                       5               ``1/2``
``weibull``                     6               ``p``
``linear`` (default)            7               ``1 - p``
``median_unbiased``             8               ``p/3 + 1/3``
``normal_unbiased``             9               ``p/4 + 3/8``
=============================== =============== ===============

Note that indices j and j + 1 are clipped to the range 0 to n - 1 when the results of the formula would be outside the allowed range of non-negative indices, and resulting p is clipped to the range 0 to 1.

The table above includes only the estimators from ^[1] that are continuous functions of probability p (estimators 4-9). SciPy also provides the three discontinuous estimators from ^[1] (estimators 1-3), where j is defined as above, m is defined as follows, and g is 0 when index = p*n + m - 1 is less than 0 and otherwise is defined below.

• inverted_cdf: m = 0 and g = int(index - j > 0)

• averaged_inverted_cdf: m = 0 and g = (1 + int(index - j > 0)) / 2

• closest_observation: m = -1/2 and g = 1 - int((index == j) & (j%2 == 1))

When all the data in x are unique, estimated_cdf and quantile are are inverses of one another within a certain domain. Although quantile with method='linear' is invertible over the whole domain of p from 0 to 1, this is not true of other methods.

Array API Standard Support

estimated_cdf 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                   ✅                     ✅                   
Dask                  ⛔                     n/a                 
====================  ====================  ====================

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

See dev-arrayapi for more information.