Given two continuous distributions $\mathcal{F}_X$ and $\mathcal{F}_Y$, It is not clear to me whether the relation of convex dominance among them:
$$(0)\quad \mathcal{F}_X <_c \mathcal{F}_Y$$
implies that
$$(1)\quad F_Y^{-1}(q) \leq F_X^{-1}(q),\quad \forall q\in[0.5,1]$$
holds or if some further hypothesis are needed if $(1)$ is to hold?
Definition of Convex dominance.
If two continuous distributions $\mathcal{F}_X$ and $\mathcal{F}_Y$ satisfy:
$$(2)\quad F_Y^{-1}F_X(x)\text{ is convex in } x$$
[0] then we write:
$$F_X <_c F_Y$$
and say that $\mathcal{F}_Y$ is more right skewed than $\mathcal{F}_X$. Because $F_X$ and $F_Y$ are probability distributions, $(2)$ also implies that the derivative of $F_Y^{-1}F_X(x)$ is monotonically non decreasing and non-negative [1], that $F_Y^{-1}F_X(x)-x$ is convex [2], that $F_X$ and $F_{aY+b}$ cross each other at most twice $\forall a>0,b\in\mathbb{R}$ [2] and that [2], for $\forall p\in[0,0.5]$:
$$\frac{F^{-1}_X(p)}{F^{-1}_Y(p)}\geq\frac{F^{-1}_X(1-p)}{F^{-1}_Y(1-p)}.$$
- [0] Zwet, W.R. van (1964). Convex Transformations of Random Variable. (1964). Amterdam: Mathematish Centrum.
- [1] Oja, H. (1981). On Location, Scale, Skewness and Kurtosis of Univariate Distributions. Scandinavian Journal of Statistics. Vol. 8, pp. 154--168
- [2] R.A. Groeneveld and G. Meeden. (1984). Measuring skewness and kurtosis. The Statistician. 33:391-399.