# 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 𝔼[ϕ(Y_1)] ≤𝔼[ϕ(Y_2)] for all increasing concave functions ϕ for which the expected values exist, and smaller than Y_2 in the increasing convex order if 𝔼[ψ(Y_1)] ≤𝔼[ψ(Y_2)] for all increasing convex ψ. This article develops nonparametric estimators for the conditional cumulative distribution functions F_x(y) = ℙ(Y ≤ y | 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, …, K} and for continuously distributed X.

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