  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.
