Consistent Estimation of Distribution Functions under Increasing Concave and Convex Stochastic Ordering

A random variable <i>Y</i>₁ is said to be smaller than <i>Y</i>₂ in the increasing concave stochastic order if E[ϕ(Y1)]≤E[ϕ(Y2)] for all increasing concave functions ϕ for which the expected values exist, and smaller than <i>Y</i>₂ in the increasing convex order if E[ψ(Y1)]≤E[ψ(Y2)] for all increasing convex <i>ψ</i>. This article develops nonparametric estimators for the conditional cumulative distribution functions Fx(y)=ℙ(Y≤y|X=x) of a response variable <i>Y</i> given a covariate <i>X</i>, solely under the assumption that the conditional distributions are increasing in <i>x</i> in the increasing concave or increasing convex order. Uniform consistency and rates of convergence are established both for the <i>K</i>-sample case X∈{1,…,K} and for continuously distributed <i>X</i>.