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

2021/05/07 by Alexander Henzi, Henzi, Alexander · 1 citation
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2105.03101

openalex publication_date 2021/05/07 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

A random variable Y₁ is said to be smaller than Y₂ in the increasing concave stochastic order if 𝔼[ϕ(Y₁)] ≤ 𝔼[ϕ(Y₂)] for all increasing concave functions ϕ for which the expected values exist, and smaller than Y₂ in the increasing convex order if 𝔼[ψ(Y₁)] ≤ 𝔼[ψ(Y₂)] for all increasing convex ψ. 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 ∈ \1, …, K\ and for continuously distributed X.

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