Consistent Estimation of Distribution Functions under Increasing Concave and Convex Stochastic Ordering

Abstract

A random variable Y₁ is said to be smaller than Y₂ in the increasing concave stochastic order if $\mathbb{E}[\varphi (Y_1)]\le \mathbb{E}[\varphi (Y_2)]$ for all increasing concave functions $\varphi$ for which the expected values exist, and smaller than Y₂ in the increasing convex order if $\mathbb{E}[\psi (Y_1)]\le \mathbb{E}[\psi (Y_2)]$ for all increasing convex ψ. This article develops nonparametric estimators for the conditional cumulative distribution functions $F_x(y)=\mathrm{\mathbb{P}}(Y\le y|X=x)$ 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 $X\in \{1,\dots ,K\}$ and for continuously distributed X.

Keywords:

• Conditional distribution estimation
• Second-order stochastic dominance
• Stochastic order
• Uniform consistency

Supplementary Materials

Proofs and additional figures:Proofs of the theoretical results in Sections 2 and 3, and additional figures for the simulation study in Section 5. (.pdf file)

Code and replication material:R code implementing the estimators, and replication material for the simulations and case study. (.zip file)

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

The author is grateful to Johanna Ziegel and Timo Dimitriadis for helpful comments and discussions. Many useful comments by the editor, the associate editor and anonymous referees have helped to improve this article.

Additional information

Funding

This work was supported by the Swiss National Science Foundation.