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

TL;DR

This paper develops nonparametric estimators for conditional distribution functions under increasing concave and convex stochastic orders, proving their uniform consistency and convergence rates for both discrete and continuous covariates.

Contribution

It introduces novel nonparametric estimators for conditional distributions constrained by stochastic orders, with theoretical guarantees.

Findings

Establishes uniform consistency of the estimators.

Derives convergence rates for discrete and continuous covariates.

Provides theoretical validation for the proposed estimators.

Abstract

A random variable $Y_1$ is said to be smaller than $Y_2$ in the increasing concave stochastic order if $\mathbb{E}[\phi(Y_1)] \leq \mathbb{E}[\phi(Y_2)]$ for all increasing concave functions $\phi$ for which the expected values exist, and smaller than $Y_2$ in the increasing convex order if $\mathbb{E}[\psi(Y_1)] \leq \mathbb{E}[\psi(Y_2)]$ for all increasing convex $\psi$. This article develops nonparametric estimators for the conditional cumulative distribution functions $F_x(y) = \mathbb{P}(Y \leq y \mid X = x)$ 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 $X \in \{1, \dots, K\}$ and for continuously distributed $X$.