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

arXiv: https://arxiv.org/abs/2105.03101 · HTML: https://mlsanity.com/paper/2105.03101/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 May 2021.

## Built on by (most cited first, 1 of 1)

- [Risk Bounds For Distributional Regression](https://mlsanity.com/paper/2505.09075/) (2025)

## Abstract (arXiv)

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.