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Register a new userGaussian linear approximation for the estimation of the Shapley effects
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Added by
Baptiste Broto
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
Authors
Baptiste Broto
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In this paper, we address the estimation of the sensitivity indices called Shapley eects. These sensitivity indices enable to handle dependent input variables. The Shapley eects are generally dicult to estimate, but they are easily computable in the Gaussian linear framework. The aim of this work is to use the values of the Shapley eects in an approximated Gaussian linear framework as estimators of the true Shapley eects corresponding to a non-linear model. First, we assume that the input variables are Gaussian with small variances. We provide rates of convergence of the estimated Shapley eects to the true Shapley eects. Then, we focus on the case where the inputs are given by an non-Gaussian empirical mean. We prove that, under some mild assumptions, when the number of terms in the empirical mean increases, the dierence between the true Shapley eects and the estimated Shapley eects given by the Gaussian linear approximation converges to 0. Our theoretical results are supported by numerical studies, showing that the Gaussian linear approximation is accurate and enables to decrease the computational time signicantly.
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Let $X^{(n)}$ be an observation sampled from a distribution $P_{theta}^{(n)}$ with an unknown parameter $theta,$ $theta$ being a vector in a Banach space $E$ (most often, a high-dimensional space of dimension $d$). We study the problem of estimation of $f(theta)$ for a functional $f:Emapsto {mathbb R}$ of some smoothness $s>0$ based on an observation $X^{(n)}sim P_{theta}^{(n)}.$ Assuming that there exists an estimator $hat theta_n=hat theta_n(X^{(n)})$ of parameter $theta$ such that $sqrt{n}(hat theta_n-theta)$ is sufficiently close in distribution to a mean zero Gaussian random vector in $E,$ we construct a functional $g:Emapsto {mathbb R}$ such that $g(hat theta_n)$ is an asymptotically normal estimator of $f(theta)$ with $sqrt{n}$ rate provided that $s>frac{1}{1-alpha}$ and $dleq n^{alpha}$ for some $alphain (0,1).$ We also derive general upper bounds on Orlicz norm error rates for estimator $g(hat theta)$ depending on smoothness $s,$ dimension $d,$ sample size $n$ and the accuracy of normal approximation of $sqrt{n}(hat theta_n-theta).$ In particular, this approach yields asymptotically efficient estimators in some high-dimensional exponential models.
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