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Module « scipy.stats »

Fonction kstat - module scipy.stats

Signature de la fonction kstat

def kstat(data, n=2, *, axis=None, nan_policy='propagate', keepdims=False) 

Description

help(scipy.stats.kstat)

    


Return the `n` th k-statistic ( ``1<=n<=4`` so far).

The `n` th k-statistic ``k_n`` is the unique symmetric unbiased estimator of the
`n` th cumulant :math:`\kappa_n` [1]_ [2]_.

Parameters
----------
data : array_like
    Input array.
n : int, {1, 2, 3, 4}, optional
    Default is equal to 2.
axis : int or None, default: None
    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
-------
kstat : float
    The `n` th k-statistic.

See Also
--------

:func:`kstatvar`
    Returns an unbiased estimator of the variance of the k-statistic
:func:`moment`
    Returns the n-th central moment about the mean for a sample.


Notes
-----
For a sample size :math:`n`, the first few k-statistics are given by

.. math::

    k_1 &= \frac{S_1}{n}, \\
    k_2 &= \frac{nS_2 - S_1^2}{n(n-1)}, \\
    k_3 &= \frac{2S_1^3 - 3nS_1S_2 + n^2S_3}{n(n-1)(n-2)}, \\
    k_4 &= \frac{-6S_1^4 + 12nS_1^2S_2 - 3n(n-1)S_2^2 - 4n(n+1)S_1S_3
    + n^2(n+1)S_4}{n (n-1)(n-2)(n-3)},

where

.. math::

    S_r \equiv \sum_{i=1}^n X_i^r,

and :math:`X_i` is the :math:`i` th data point.

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
----------
.. [1] http://mathworld.wolfram.com/k-Statistic.html

.. [2] http://mathworld.wolfram.com/Cumulant.html

Examples
--------
>>> from scipy import stats
>>> from numpy.random import default_rng
>>> rng = default_rng()

As sample size increases, `n`-th moment and `n`-th k-statistic converge to the
same number (although they aren't identical). In the case of the normal
distribution, they converge to zero.

>>> for i in range(2,8):
...     x = rng.normal(size=10**i)
...     m, k = stats.moment(x, 3), stats.kstat(x, 3)
...     print(f"{i=}: {m=:.3g}, {k=:.3g}, {(m-k)=:.3g}")
i=2: m=-0.631, k=-0.651, (m-k)=0.0194  # random
i=3: m=0.0282, k=0.0283, (m-k)=-8.49e-05
i=4: m=-0.0454, k=-0.0454, (m-k)=1.36e-05
i=6: m=7.53e-05, k=7.53e-05, (m-k)=-2.26e-09
i=7: m=0.00166, k=0.00166, (m-k)=-4.99e-09
i=8: m=-2.88e-06 k=-2.88e-06, (m-k)=8.63e-13

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