On symmetry‐modulated distributions: revisiting an old result and a step further

Abstract

In the context of modulated-symmetry distributions, there exist various forms of skew-elliptical families. We present yet another one, but with an unusual feature: the modulation factor of the baseline elliptical density is represented by a distribution function with an argument that is not an odd function, as it occurs instead with the overwhelming majority of similar formulations, not only with other skew-elliptical families. The proposal is obtained by going back to the use of a lemma known since 1999, which can be seen as the general frame for a vast number of existing formulations, and use it on a different route. The broader target is to show that this "mother lemma" can still generate novel progeny. The final part of the paper examines a further level of generalization of the "mother-lemma" where symmetry conditions are removed.

A Selective Overview of Skew-Elliptical and Related Distributions and of Their Applications

“…To demonstrate the potential of Proposition 1 beyond the domain of validity of Proposition 3, Azzalini and Regoli [62] have constructed densities with baseline f 0 of EC type but without enforcing the condition w(−x) = −w(x) in (20). The resulting distributions enjoy some interesting formal properties and they can have similar flexibility of those generated from Proposition 3.…”