Conditional Mean Independence and Global Sensitivity Analysis using Nearest Neighbor Graphs

2026-07-06

AI Summary

This work proposes a nearest-neighbor-graph-based estimator for the proportion of explained variation by the conditional mean function, offering a unified framework for model evaluation, model-free variable screening, and higher-order Sobol’ interaction analysis. The proposed estimator achieves nearly linear computational complexity and enables asymptotically standard normal inference without resorting to bootstrapping or sample splitting, thereby guaranteeing correct significance levels and universal consistency. Theoretical results establish its convergence rate and asymptotic normality, while extensive simulations and real-data analyses demonstrate substantial improvements over existing methods in both computational efficiency and statistical power.

Abstract

Quantifying how well a conditional mean function explains a response is central to many statistical tasks, such as model evaluation and feature screening. A basic nonparametric measure of such dependence is the proportion of variation in the response explained by the regression function, which can also be interpreted as a multivariate Sobol' index, a fundamental notion in global sensitivity analysis. In this paper, we propose a consistent estimator of this measure based on nearest neighbor graphs that can be computed in near-linear time. We also derive its rate of convergence and show that a studentized version of the estimator is asymptotically standard normal under the null hypothesis of conditional mean independence. This leads to a computationally efficient test for conditional mean independence that attains the correct asymptotic level and is universally consistent, without requiring bootstrap calibration or sample splitting. Next, we use the proposed estimator to develop a model-free variable screening algorithm that is provably consistent. We also discuss extensions of the framework to measuring interaction effects using higher-order Sobol' indices. The benefits of the proposed methods are demonstrated through simulation studies and a real-data example.

Problem

• conditional mean independence
• global sensitivity analysis
• Sobol' index
• variable screening
• nearest neighbor graphs

Innovation

• nearest neighbor graphs
• conditional mean independence
• Sobol' index
• global sensitivity analysis
• model-free variable screening