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

A random variable Y1 is said to be smaller than Y2 in the increasing concave stochastic order if mathbbE[φ(Y1)] ≤ mathbbE[φ(Y2)] for all increasing concave functions φ for which the expected values exist, and smaller than Y2 in the increasing convex order if mathbbE[psi(Y1)] ≤ mathbbE[psi(Y2)] for all increasing convex psi. This article develops nonparametric estimators for the conditional cumulative distribution functions Fx(y) = mathbbP(Y ≤ 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 ∈ 1, dots, K and for continuously distributed X.