On the unimodality of power transformations of positive stable densities

TL;DR

This paper investigates the conditions under which power transformations of positive stable densities are unimodal, identifying a specific domain where unimodality fails based on the parameters.

Contribution

It characterizes the exact parameter domain where positive stable densities' power transformations are unimodal, revealing a cusp-shaped boundary in the parameter space.

Findings

Unimodality depends on parameters within a specific domain D.

Domain D is unbounded with a cusp at (1/2, -1/2).

Power transformations are unimodal outside domain D.

Abstract

Let $Z_\alpha$ be a positive $\alpha-$stable random variable and $r\in{\bf R}.$ We show the existence of an unbounded open domain $D$ in $[1/2,1]\times{\bf R}$ with a cusp at $(1/2,-1/2)$, characterized by the complete monotonicity of the function $F_{\alpha, r} (\lambda) = (\alpha \lambda^\alpha -r)e^{-\lambda^\alpha}/!/! ,$ such that $Z_\alpha^r$ is unimodal if and only if $(\alpha, r)\notin D.$