On the Composition of Unimodal Distributions

Abstract

A distribution function is called strong unimodal if its composition with any unimodal distribution function is unimodal. The following theorem is proved: For a proper unimodal distribution $F(x)$ to be strong unimodal, it is necessary and sufficient that the function $F(x)$ be continuous, and the function log $F'(x)$ be concave at a set of points where neither the right nor the left derivative of the function $F(x)$ is equal to zero.