`scipy.stats._morestats:kstat`

source: build-install/usr/lib/python3.14/site-packages/scipy/stats/_morestats.py:236

Signature

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

Summary

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

Extended Summary

The n th k-statistic k_n is the unique symmetric unbiased estimator of the n th cumulant $κ_{n}$ ^[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.

nan_policy='raise' and nan_policy='omit' are not supported by lazy backends, and nan_policy='omit' is not compatible with MArray input. For multidimensional input, nan_policy='omit' is supported only by the NumPy backend. Instead, mask NaN values using MArray; and use the default nan_policy. See Notes for support information.

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.

Notes

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

k_{1} k_{2} k_{3} k_{4} = \frac{S _{1}}{n}, = \frac{n S _{2} - S _{1}^{2}}{n ( n - 1 )}, = \frac{2 S _{1}^{3} - 3 n S _{1} S _{2} + n ^{2} S _{3}}{n ( n - 1 ) ( n - 2 )}, = \frac{- 6 S _{1}^{4} + 12 n S _{1}^{2} S _{2} - 3 n ( n - 1 ) S _{2}^{2} - 4 n ( n + 1 ) S _{1} S _{3} + n ^{2} ( n + 1 ) S _{4}}{n ( n - 1 ) ( n - 2 ) ( n - 3 )},

where

S_{r} \equiv i = 1 \sum n X_{i}^{r},

and $X_{i}$ is the $i$ 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.

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=-0.238, k=-0.245, (m-k)=0.00731
i=3: m=-0.0833, k=-0.0836, (m-k)=0.000251
i=4: m=0.00797, k=0.00797, (m-k)=-2.39e-06
i=5: m=0.0069, k=0.0069, (m-k)=-2.07e-07
i=6: m=-0.00264, k=-0.00264, (m-k)=7.93e-09
i=7: m=0.00035, k=0.00035, (m-k)=-1.05e-10

`scipy.stats._morestats:kstat`

Signature

Summary

Extended Summary

Parameters

Returns

Notes

Examples

See also

Aliases

Referenced by

This package