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Centre-free kurtosis orderings for asymmetric distributions

Eberl, Andreas ORCID iD icon 1; Klar, Bernhard ORCID iD icon 1
1 Institut für Stochastik (STOCH), Karlsruher Institut für Technologie (KIT)

Abstract:

The concept of kurtosis is used to describe and compare theoretical and empirical distributions in a multitude of applications. In this connection, it is commonly applied to asymmetric distributions. However, there is no rigorous mathematical foundation establishing what is meant by kurtosis of an asymmetric distribution and what is required to measure it properly. All corresponding proposals in the literature centre the comparison with respect to kurtosis around some measure of central location. Since this either disregards critical amounts of information or is too restrictive, we instead revisit a canonical approach that has barely received any attention in the literature. It reveals the non-transitivity of kurtosis orderings due to an intrinsic entanglement of kurtosis and skewness as the underlying problem. This is circumvented by restricting attention to sets of distributions with equal skewness, on which the proposed kurtosis ordering is shown to be transitive. Moreover, we introduce a functional that preserves this order for arbitrary asymmetric distributions. As application, we examine the families of Weibull and sinh-arcsinh distributions and show that the latter family exhibits a skewness-invariant kurtosis behaviour.


Verlagsausgabe §
DOI: 10.5445/IR/1000156220
Veröffentlicht am 23.02.2023
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Stochastik (STOCH)
Publikationstyp Zeitschriftenaufsatz
Publikationsjahr 2023
Sprache Englisch
Identifikator ISSN: 0932-5026, 0039-0631, 1613-9798
KITopen-ID: 1000156220
Erschienen in Statistical Papers
Verlag Springer
Vorab online veröffentlicht am 09.02.2023
Schlagwörter Asymmetric distribution, Higher-order convexity, Kurtosis, Skewness, Stochastic order, Sinh-arcsinh distribution
Nachgewiesen in Web of Science
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Scopus
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