Natural exponential families and self-decomposability

Let F be a full natural exponential family on R which is generated by a self-decomposable probability distribution P. We provide a necessary and sufficient condition on P under which all other elements of F are also self-decomposable. Moreover, we show that if F is self-decomposable (i.e., composed of self-decomposable elements), then the exponential dispersion model generated by F shares the same property. The statistical importance of such results is linked to the fact that self-decomposable distributions are absolutely continuous and unimodal, thus providing potential exponential dispersion models for modeling data stemming from absolutely continuous and unimodal populations.