The relationship between shape parameters and kurtosis in some relevant models

When a distributional model is chosen, the analytic relation between its shape parameters and the values taken by some kurtosis indexes, especially if they are unconventional, is rarely known. In addition, different indexes may provide contrasting evidence about the level of global kurtosis, when the parameters of the model are varied. That happens because just few parameters act “plainly” on kurtosis, namely so as to produce consistent modifications of the shape of the graph on both its sides. Many parameters, instead, affect kurtosis along with a change of the skewness of the distribution, that is by “inflating” a single side of the graph (usually a tail) at the expense of the other. Thanks to some relevant examples, this paper tries to provide general indications to recognize the two kinds of parameters above and to interpret their effect on the classical Pearson’s standardized fourth moment and on some lesser known kurtosis indexes. Specifically, it is shown that only a decomposed analysis of indexes can help to understand their apparent contradictions, especially when some of them are too sensitive to changes in the tails. Finally, some applications are provided.