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

Abstract

A random variable $Y_1$ is said to be smaller than $Y_2$ in the increasing concave stochastic order if $\mathbb{E}[\phi(Y_1)] \leq \mathbb{E}[\phi(Y_2)]$ for all increasing concave functions $\phi$ for which the expected values exist, and smaller than $Y_2$ in the increasing convex order if $\mathbb{E}[\psi(Y_1)] \leq \mathbb{E}[\psi(Y_2)]$ for all increasing convex $\psi$. This article develops nonparametric estimators for the conditional cumulative distribution functions $F_x(y) = \mathbb{P}(Y \leq y \mid X = x)$ 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 $X$.