A Martingale Approach To Fluctuations of Rank Estimators in Sensitivity Analysis

TL;DR

This paper develops a martingale-based framework to analyze the fluctuations of rank-based sensitivity estimators, unifying and simplifying existing results, and providing explicit formulas for asymptotic variances of Sobol' and CvM indices.

Contribution

It introduces a unified martingale approach to derive sharp fluctuation results for various sensitivity indices, including new explicit formulas for their asymptotic variances.

Findings

Unified treatment of Sobol' and CvM indices fluctuations.

Explicit asymptotic variance formulas for sensitivity estimators.

Simplification and unification of previous fluctuation results.

Abstract

Given a bivariate random pair $(X,Y)$, a natural problem is to estimate, from a single sample $(X_i,Y_i)_{1\le i\le n}$, quantities such as $\mathbb{E}\left[ \mathbb{E}[ Y\mid X ]^2 \right]$. More broadly, sensitivity indices are designed to quantify the possibly nonlinear influence of an input variable $X$ on an output variable $Y$. A classical example is the Sobol' index $$ \frac{\mathrm{Var}(\mathbb{E}[Y\mid X])}{\mathrm{Var}(Y)} \in [0,1] \ . $$ Another important example is the Cram\'er--von Mises (CvM) index. Following the pioneering work of Chatterjee \cite{chatterjee2021new}, consistent rank-based estimators are now available for such quantities. In this paper, we prove sharp fluctuation results using martingale methods. Our framework yields a unified treatment of the univariate Sobol' index, a multivariate extension involving several functions of the same scalar input, and the CvM index. As a consequence, we recover, unify, and simplify results from Gamboa et al. \cite{gamboa2022global, gamboa2023erratum}, Lin--Han \cite{lin2022limit}, and Kroll \cite{kroll2024asymptotic}. In particular, we work under minimal regularity assumptions. Furthermore, while the Gaussian fluctuation phenomenon itself was already known, the novelty lies in the structure of the asymptotic variance: for the CvM index, we obtain, to the best of our knowledge, the first explicit formula, while for the Sobol' index, we derive a new expression with a more structured form.