Given a CDF $F:\mathbb{R}_{\geq 0}\to [0,1]$ such that $F(\alpha x) \geq \beta F(x)$ for some $1>\alpha,\beta>0.$ I'm trying to figure out if there is any known family of distributions for which this condition holds. Is this somehow related to log-concavity? Is there any connection to reverse hazard rate?
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Trivial example: $X\sim\operatorname{Uniform}(0,a)$ with $a>0$. Then $F_X(x)=x$. It follows for $\alpha>1$ and $\beta>0$ that $F_X(\alpha x)\geq \beta F_X(x)$ whenever $\alpha\geq\beta$. – Aaron Hendrickson Jul 01 '22 at 13:16
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Thanks a lot! I'm actually looking for something more general than examples, rather a known condition (like log-concavity, regularity, MHR etc.) or at least the connection to such families – Tal Alon Jul 01 '22 at 13:21
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@Aaron Hendrickson You forgot what happens for $x>a$. @ Tal Alon If $\alpha > 1 \ge \beta > 0$, the inequality holds for all distribution function on $\mathbb{R}_+$, but it is not very interesting. If $\beta>1$, it cannot hold for all $x \ge 0$, because of the limit of $F$ at $+\infty$ is $1$. – Christophe Leuridan Jul 01 '22 at 14:14