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

  • y (array-like) –
  • axis (int or None, optional) – Axis along which the kurtoses are computed. If None, the entire array is used.
  • ab (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
  • db (iterable, 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
  • excess (bool, optional) – If true (default), computed values are excess of those for a standard normal distribution.
  • kr1 (ndarray) – The standard kurtosis estimator.
  • kr2 (ndarray) – Kurtosis estimator based on octiles.
  • kr3 (ndarray) – Kurtosis estimators based on exceedence expectations.
  • kr4 (ndarray) – 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.

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