Shapley Curves: A Smoothing Perspective

TL;DR

This paper introduces Shapley curves to understand variable importance from a nonparametric perspective, providing theoretical analysis and a bootstrap method for inference, with applications to vehicle price factors.

Contribution

It defines population-level Shapley curves, derives convergence rates and asymptotic normality, and proposes a bootstrap method for finite sample inference.

Findings

Theoretical convergence rates for Shapley curve estimators

Asymptotic normality results under general conditions

Effective bootstrap procedure for inference

Abstract

This paper fills the limited statistical understanding of Shapley values as a variable importance measure from a nonparametric (or smoothing) perspective. We introduce population-level \textit{Shapley curves} to measure the true variable importance, determined by the conditional expectation function and the distribution of covariates. Having defined the estimand, we derive minimax convergence rates and asymptotic normality under general conditions for the two leading estimation strategies. For finite sample inference, we propose a novel version of the wild bootstrap procedure tailored for capturing lower-order terms in the estimation of Shapley curves. Numerical studies confirm our theoretical findings, and an empirical application analyzes the determining factors of vehicle prices.