My musings on a pioneering work of Erich Lehmann and its rediscoveries on some families of distributions

Abstract

The paper of Erich Lehmann (Lehmann 1953) is renowned for its ground-breaking contribution to rank tests within the area of nonparametric statistics. This work, in all likelihood, is known to every statistician. But, what is likely not known to many are some of the novel concepts and models that this paper succinctly introduced to the area of distribution theory. This, unfortunately, has led to some of these being rediscovered in the literature and then being referred to under different names. The purpose of this note is, therefore, two-fold: first to explain the key models that are contained in the mentioned work of Erich Lehmann, and second to point out how some of the known models discussed in the distribution theory and stochastic modeling literature are indeed present either explicitly or implicitly in the paper of Lehmann.

Keywords:

• Lehmann families
• exponentiated distributions
• transmuted distributions
• proportional hazards model
• proportional reversed hazards model

Acknowledgments

The author thanks the Natural Sciences and Engineering Research Council of Canada for funding this research through an Individual Discovery Grant 2020-06733, and the Associate Editor and the Reviewer for some useful comments and suggestions on an earlier version of this manuscript.