[Paper Review] A contamination model for approximate stochastic order: extended version

Pedro C. Álvarez-Esteban, Eustasio del Barrio+2 more|arXiv (Cornell University)|Dec 5, 2014

Statistical Distribution Estimation and Applications35 references3 citations

TL;DR

This paper proposes a contamination model to assess approximate stochastic order between two distributions, where one distribution is a small fraction of contaminated data from another. By modeling distributions as mixtures with a small contamination level 𝜋, the method quantifies deviation from exact stochastic order and provides asymptotic inference for the minimal contamination level, offering a flexible alternative to rigid stochastic dominance testing.

ABSTRACT

Stochastic ordering among distributions has been considered in a variety of scenarios. Economic studies often involve research about the ordering of investment strategies or social welfare. However, as noted in the literature, stochastic orderings are often a too strong assumption which is not supported by the data even in cases in which the researcher tends to believe that a certain variable is somehow smaller than other. Instead of considering this rigid model of stochastic order we propose to look at a more flexible version in which two distributions are said to satisfy an approximate stochastic order relation if they are slightly contaminated versions of distributions which do satisfy the stochastic ordering. The minimal level of contamination that makes this approximate model hold can be used as a measure of the deviation of the original distributions from the exact stochastic order model. Our approach is based on the use of trimmings of probability measures. We discuss the connection between them and the approximate stochastic order model and provide theoretical support for its use in data analysis. We also provide simulation results.

Motivation & Objective

• To address the rigidity of exact stochastic order, which often fails to hold in practice despite intuitive expectations of stochastic dominance.
• To develop a flexible statistical model that allows for small deviations from stochastic order through contamination of distributions.
• To quantify the minimal contamination level 𝜋 required to make two distributions satisfy stochastic order, serving as a measure of deviation from the exact model.
• To provide asymptotic theory and inference tools for estimating and testing this minimal contamination level in finite samples.
• To offer a practical alternative to traditional stochastic dominance testing, especially in cases where exact order is implausible but approximate order is plausible.

Proposed method

• Proposes a contamination model where distribution F is modeled as F = (1−𝜋)F̃ + 𝜋H, with F̃ ≤st G, allowing F to be a small fraction of contamination from H.
• Uses trimmings of probability measures to formalize the idea of removing or adjusting extreme or anomalous observations to achieve stochastic order.
• Defines the minimal contamination level 𝜋(F,G) as the smallest 𝜋 such that the contamination model holds, serving as a measure of deviation from exact stochastic order.
• Develops asymptotic distribution theory for the estimator 𝜋̂n,n of the minimal contamination level, showing convergence to a functional of Brownian bridges.
• Derives a pivotal asymptotic distribution for inference on 𝜋(F,G), enabling confidence intervals and hypothesis tests via critical values K1−𝛼(𝜋0, 1/2).
• Employs empirical process theory and oscillation bounds (e.g., Stute’s results) to control error terms in the asymptotic analysis of the contamination estimator.

Experimental results

Research questions

• RQ1 How can we measure the degree to which two distributions deviate from exact stochastic order in a statistically meaningful way?
• RQ2 What is the minimal level of contamination required to transform two non-stochastically ordered distributions into a stochastically ordered pair?
• RQ3 Can we construct valid asymptotic confidence intervals for the minimal contamination level under the approximate stochastic order model?
• RQ4 How does the asymptotic distribution of the contamination estimator behave under the null hypothesis of stochastic dominance?
• RQ5 What are the finite-sample properties of the contamination-based inference procedure, and how does it compare to standard stochastic dominance tests?

Key findings

• The minimal contamination level 𝜋(F,G) provides a quantitative measure of deviation from exact stochastic order, with smaller values indicating closer agreement with the order model.
• The estimator 𝜋̂n,n of the minimal contamination level is asymptotically normal, with √(n/2)(𝜋̂n,n − 𝜋0) converging in distribution to a functional of Brownian bridges.
• The asymptotic distribution of the estimator is pivotal and depends only on the true contamination level 𝜋0 and the uniform bound 1/2, enabling critical value computation.
• The method ensures asymptotic validity of hypothesis tests and confidence intervals for the contamination level, even under weak regularity conditions.
• Simulation results and a case study demonstrate the method’s practical utility in detecting approximate stochastic order where exact order fails.
• Theoretical analysis confirms that the estimator is consistent and that the limiting distribution is continuous in the parameter space, supporting robust inference.