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 for all increasing concave functions for which the expected values exist, and smaller than Y 2 in the increasing convex order if for all increasing convex ψ. This article develops nonparametric estimators for the conditional cumulative distribution functions 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 and for continuously distributed X.