On the unimodality of power transformations of positive stable densities

Abstract

Let $Z_{\alpha }$ be a positive $\alpha -$stable random variable and $r\in \mathbf{R}.$ We show the existence of an unbounded open domain $D$ in $[1/2,1]\times \mathbf{R}$ 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 $(\alpha ,r)\notin D.$

References in corpus (1)

• A refined factorization of the exponential law

Cited by in corpus (2)

• Further examples of GGC and HCM densities
• Comparing Fréchet and positive stable laws