scipy.stats.VonMises.

kurtosis#

VonMises.kurtosis(*, method=None, convention=None)[source]#

Kurtosis, a measure of tailedness

For real-line distributions, the skewness is the standardized fourth moment.

The kurtosis \(k\) of a circular random variable \(X\) is defined ([2] 31) as

\[k = \frac{\bar{\alpha}_2 - \rho^2} { (1 - \rho)^{2} }\]

where \(\rho\) and \(\bar{\alpha}_2\) are the real components of the first and second central trigonometric moments of \(X\), respectively.

Parameters:
method{None, ‘formula’, ‘general’, ‘transform’, ‘normalize’, ‘cache’}

Method used to calculate the standardized fourth moment. Not all methods are available for all distributions. See moment for details.

conventionstr, optional

For real-line distributions, two distinct conventions are available:

  • 'non-excess': the standardized fourth moment (Pearson’s kurtosis)

  • 'excess': the standardized fourth moment minus 3 (Fisher’s kurtosis)

The default is 'non-excess'.

For circular distributions, only the default convention [2] is supported.

See also

moment
mean
variance

Notes

By default, this is the standardized fourth moment, also known as the “non-excess” or “Pearson” kurtosis (e.g. the kurtosis of the normal distribution is 3). The “excess” or “Fisher” kurtosis (the standardized fourth moment minus 3) is available via the convention parameter.

References

[1]

Kurtosis, Wikipedia, https://en.wikipedia.org/wiki/Kurtosis

[2] (1,2)

Mardia, Kanti V., and Peter E. Jupp. Directional statistics. John Wiley & Sons, 1999. DOI:10.1002/9780470316979.

Examples

Instantiate a distribution with the desired parameters:

>>> from scipy import stats
>>> X = stats.Normal(mu=1., sigma=2.)

Evaluate the kurtosis:

>>> X.kurtosis()
3.0
>>> (X.kurtosis()
...  == X.kurtosis(convention='excess') + 3.
...  == X.moment(order=4, kind='standardized'))
True
On this page