statsmodels.stats.stattools.robust_kurtosis(y, axis=0, ab=(5.0, 50.0), dg=(2.5, 25.0), excess=True)[source]

Calculates the four kurtosis measures in Kim & White


Data to compute use in the estimator.

axisint or None, optional

Axis along which the kurtosis are computed. If None, the entire array is used.

a iterable, optional

Contains 100*(alpha, beta) in the kr3 measure where alpha is the tail quantile cut-off for measuring the extreme tail and beta is the central quantile cutoff for the standardization of the measure

dbiterable, optional

Contains 100*(delta, gamma) in the kr4 measure where delta is the tail quantile for measuring extreme values and gamma is the central quantile used in the the standardization of the measure

excessbool, optional

If true (default), computed values are excess of those for a standard normal distribution.


The standard kurtosis estimator.


Kurtosis estimator based on octiles.


Kurtosis estimators based on exceedance expectations.


Kurtosis measure based on the spread between high and low quantiles.


The robust kurtosis measures are defined

\[KR_{2}=\frac{\left(\hat{q}_{.875}-\hat{q}_{.625}\right) +\left(\hat{q}_{.375}-\hat{q}_{.125}\right)} {\hat{q}_{.75}-\hat{q}_{.25}}\]
\[KR_{3}=\frac{\hat{E}\left(y|y>\hat{q}_{1-\alpha}\right) -\hat{E}\left(y|y<\hat{q}_{\alpha}\right)} {\hat{E}\left(y|y>\hat{q}_{1-\beta}\right) -\hat{E}\left(y|y<\hat{q}_{\beta}\right)}\]
\[KR_{4}=\frac{\hat{q}_{1-\delta}-\hat{q}_{\delta}} {\hat{q}_{1-\gamma}-\hat{q}_{\gamma}}\]

where \(\hat{q}_{p}\) is the estimated quantile at \(p\).


Tae-Hwan Kim and Halbert White, “On more robust estimation of skewness and kurtosis,” Finance Research Letters, vol. 1, pp. 56-73, March 2004.