A note on rescalings, reparametrizations and classes of distributions

Families of distributions on the real line have location, scale and shape parameters. Reparametrizations of such distributions, particularly involving rescalings, do not produce different families of distributions. However, from time to time, ‘extended’ or ‘generalized’ distributions are proposed that are nothing other than such reparametrizations. This note was triggered by a specific instance recently published in this journal but other examples are briefly mentioned as well.