Module « scipy.stats »

Fonction kstat - module scipy.stats

Signature de la fonction kstat

def kstat(data, n=2) 

Description

kstat.__doc__

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

    The nth k-statistic k_n is the unique symmetric unbiased estimator of the
    nth cumulant kappa_n.

    Parameters
    ----------
    data : array_like
        Input array. Note that n-D input gets flattened.
    n : int, {1, 2, 3, 4}, optional
        Default is equal to 2.

    Returns
    -------
    kstat : float
        The nth k-statistic.

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

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

    .. math::

        k_{1} = \mu
        k_{2} = \frac{n}{n-1} m_{2}
        k_{3} = \frac{ n^{2} } {(n-1) (n-2)} m_{3}
        k_{4} = \frac{ n^{2} [(n + 1)m_{4} - 3(n - 1) m^2_{2}]} {(n-1) (n-2) (n-3)}

    where :math:`\mu` is the sample mean, :math:`m_2` is the sample
    variance, and :math:`m_i` is the i-th sample central moment.

    References
    ----------
    http://mathworld.wolfram.com/k-Statistic.html

    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 n in [2, 3, 4, 5, 6, 7]:
    ...     x = rng.normal(size=10**n)
    ...     m, k = stats.moment(x, 3), stats.kstat(x, 3)
    ...     print("%.3g %.3g %.3g" % (m, k, m-k))
    -0.631 -0.651 0.0194  # random
    0.0282 0.0283 -8.49e-05
    -0.0454 -0.0454 1.36e-05
    7.53e-05 7.53e-05 -2.26e-09
    0.00166 0.00166 -4.99e-09
    -2.88e-06 -2.88e-06 8.63e-13