Inequalities of $s$-Ordered Distributions

Abstract

Let $C$ be the cone of functions $\phi$ that are concave-convex about the origin, continuous at the origin, and have $\phi(0) = 0$, and $\phi'(t) \leqq \phi' (-t)$ for $t \geqq 0$. Necessary and sufficient conditions are given for $\phi(\int x(t) dH(t)) \leqq \int \phi(x(t)) dH(t)$ to hold for all $\phi \in C$ and all increasing functions $x$, with $x(0) = 0$. This inequality is used to develop comparisons (i) between combinations of order statistics, and (ii) between combinations of the expected values of the order statistics, arising from distributions $F$ and $G$ in the case that $G^{-1} F \in C$. If $F(0) = G(0) = \frac{1}{2}$ and the inequality on the gradient of $F^{-1} F, (G^{-1} F)' (x) \leqq (G^{1-} F)'(-x)$ for $x > 0$, is satisfied, then $G^{-1} F \in C$ implies $F <_s G$. The inequalities presented preserve the ordering. A weaker ordering of distributions, called $r$-ordering, is defined: $F <_r G$ if and only if $F(0) = G(0) = \frac{1}{2}$ and $G^{-1} F(x)/x$ is increasing (decreasing) for $x$ positive (negative) on the support of $F$. For symmetric $r$-ordered distributions, the ratio of the expected values of the order statistics preserve the ordering.