The main objective of this paper is to estimate optimally Sobol' indices at any order when a unique input/output i.i.d.\ sample is available. Our approach stands on three main ingredients: semi-parametric estimation theory, high-order kernel estimation (inspired by the paper of Doksum in 1995), and mirror-type transformations as introduced in Bertin 2020 and Pujol 2022. We propose two different estimators. We prove that these estimators are asymptotically normal and efficient. Furthermore, we illustrate their numerical properties on standard examples.

翻译：本文的主要目标是在仅有一个独立同分布的输入/输出样本可用时，以最优方式估计任意阶的Sobol'指数。我们的方法基于三个核心要素：半参数估计理论、高阶核估计（受Doksum于1995年论文的启发），以及Bertin（2020）与Pujol（2022）提出的镜像型变换。我们提出了两种不同的估计量，并证明了它们具有渐近正态性和有效性。此外，我们通过标准算例展示了其数值特性。