Testing Conditional Stochastic Dominance at Target Points

2025-03-18

AI Summary

This paper addresses the fundamental problem of testing conditional stochastic dominance (CSD) at a given covariate value. We propose a novel nonparametric Kolmogorov–Smirnov–type test that employs data-independent critical values—eliminating the need for resampling or kernel smoothing—and introduces a support-aware refined critical value to substantially improve finite-sample power, especially for discrete data. Our unified framework, built upon induced order statistics, accommodates both continuous and discrete covariates and integrates asymptotic theory with permutation principles. We establish rigorous Type-I error control under weak regularity conditions. Monte Carlo simulations demonstrate excellent small-sample performance, and the test converges to the unconditional dominance test in limiting cases. The method is directly applicable to empirical settings including income inequality analysis, causal treatment effect evaluation, and discrimination detection.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper introduces a novel test for conditional stochastic dominance (CSD) at specific values of the conditioning covariates, referred to as target points. The test is relevant for analyzing income inequality, evaluating treatment effects, and studying discrimination. We propose a Kolmogorov--Smirnov-type test statistic that utilizes induced order statistics from independent samples. Notably, the test features a data-independent critical value, eliminating the need for resampling techniques such as the bootstrap. Our approach avoids kernel smoothing and parametric assumptions, instead relying on a tuning parameter to select relevant observations. We establish the asymptotic properties of our test, showing that the induced order statistics converge to independent draws from the true conditional distributions and that the test is asymptotically of level $alpha$ under weak regularity conditions. While our results apply to both continuous and discrete data, in the discrete case, the critical value only provides a valid upper bound. To address this, we propose a refined critical value that significantly enhances power, requiring only knowledge of the support size of the distributions. Additionally, we analyze the test's behavior in the limit experiment, demonstrating that it reduces to a problem analogous to testing unconditional stochastic dominance in finite samples. This framework allows us to prove the validity of permutation-based tests for stochastic dominance when the random variables are continuous. Monte Carlo simulations confirm the strong finite-sample performance of our method.

Problem

• Tests conditional stochastic dominance at specific covariate values
• Evaluates income inequality, treatment effects, and discrimination
• Avoids kernel smoothing and parametric assumptions

Innovation

• Kolmogorov-Smirnov test with induced order statistics
• Data-independent critical value avoids resampling
• Tuning parameter selects relevant observations