statsmodels.stats.stattools.robust_kurtosis¶

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

Parameters:¶

yarray_like

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.

Returns:¶

kr1ndarray

The standard kurtosis estimator.

kr2ndarray

Kurtosis estimator based on octiles.

kr3ndarray

Kurtosis estimators based on exceedance expectations.

kr4ndarray

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

Notes

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 \(p\).