Another Generalization of Unimodality

TL;DR

This paper introduces a new generalization of unimodality, providing simpler proofs and characterizations for certain distributions, and demonstrating the power of elementary methods over advanced techniques.

Contribution

It presents a novel generalization of unimodality, offering new proofs and characterizations, expanding on previous multivariate extensions.

Findings

Simpler proof of a conjecture on unimodality

Characterization of generalized arcsin distributions

Elementary methods outperform some advanced techniques

Abstract

A general characterization for \alpha-unimodal distributions was provided by Alamatsaz (1985) who later introduced a multivariate extension of them (Alamatsaz 1993). Here, by solving the related equations, another generalization for unimodality is presented. As a result of this generalization, a simpler proof of a conjecture, as well as a characterization for generalized arcsin distributions and some generalizations of the author's earlier works, have been obtained. Last, but not the least, it is shown that some elementary methods can be more powerful than some more advanced techniques.