Does the partial order below (R) has a name? Has anyone seem something somewhat similar or related before? The book Stochastic Orders (my bible for this stuff) has no reference on it. Any references will be extremely welcomed.
This showed up in my research and I wonder if someone saw this before. Consider two random variables $X$ and $Y$ with cumulative distribution functions, $F_X$ and $F_Y$. Let $\overline y$ be such that $F_Y(\overline y)=1$ and $F_Y(\overline y -\varepsilon)<1$ for any $\varepsilon>0$ (that is, $\overline y$ is the supremum of the support of $Y$).
We can define a partial order of random variables by: \begin{equation} X\succ_{\lambda} Y \Leftrightarrow \sup_{x\,\in\, \mathrm{supp} \,X}\dfrac{F_X(x)}{x}<\lambda\cdot \dfrac{1}{\overline y}\text{ for some fixed }\lambda\in (\frac12,1).\tag{R}\end{equation}
Obviously we can use the above rank (R) for random variables with different supports (at least we need $\overline x > \frac{\overline y}{\lambda}$.