3.11.20.2.6. 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:

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.

Returns:

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.

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 $p$.

[R84]

Tae-Hwan Kim and Halbert White, “On more robust estimation of

skewness and kurtosis,” Finance Research Letters, vol. 1, pp. 56-73, March 2004.