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[phi(Y-1)] <= 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 E[psi(Y-1)]<= E[psi(Y-2)] for all increasing convex psi. 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 is an element of{1, horizontal ellipsis ,K} and for continuously distributed X.