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. Copyright © 2018 John Wiley & Sons, Ltd.