Inference via the Skewness-Kurtosis Set

Kurtosis minus squared skewness is bounded from below by 1, but for unimodal distributions this parameter is bounded by 189/125. In some applications it is natural to compare distributions by comparing their kurtosis-minus-squared-skewness parameters. The asymptotic behavior of the empirical version of this parameter is studied here for i.i.d. random variables. The result may be used to test the hypothesis of unimodality against the alternative that the kurtosis-minus-squared-skewness parameter is less than 189/125. However, such a test has to be applied with care, since this parameter can take arbitrarily large values, also for multimodal distributions. Numerical results are presented and for three classes of distributions the skewness-kurtosis sets are described in detail.