Consistent estimation of distribution functions under increasing concave and convex stochastic ordering

A random variable $Y_1$ is said to be smaller than $Y_2$ in the increasing concave stochastic order if $\mathbb{E}[\varphi (Y_1)]\le \mathbb{E}[\varphi (Y_2)]$ for all increasing concave functions $\varphi$ for which the expected values exist, and smaller than $Y_2$ in the increasing convex order if $\mathbb{E}[\psi (Y_1)]\le \mathbb{E}[\psi (Y_2)]$ for all increasing convex $\psi$. This article develops nonparametric estimators for the conditional cumulative distribution functions $F_x(y)=\mathbb{P}(Y\le y\mid X=x)$ of a response variable $Y$ given a covariate $X$, solely under the assumption that the conditional distributions are increasing in $x$ in the increasing concave or increasing convex order. Uniform consistency and rates of convergence are established both for the $K$-sample case $X\in \{1,\dots ,K\}$ and for continuously distributed $X$.