Summary: 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.}