A Martingale Approach To Fluctuations of Rank Estimators in Sensitivity Analysis
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arXiv
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| Format: | Preprint |
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2026
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| author | Chhaibi, Reda Gamboa, Fabrice Pellegrini, Clément |
| author_facet | Chhaibi, Reda Gamboa, Fabrice Pellegrini, Clément |
| contents | 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 Cramér--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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22611 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Martingale Approach To Fluctuations of Rank Estimators in Sensitivity Analysis Chhaibi, Reda Gamboa, Fabrice Pellegrini, Clément Statistics Theory Probability 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 Cramér--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. |
| title | A Martingale Approach To Fluctuations of Rank Estimators in Sensitivity Analysis |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2603.22611 |