In the context of computer code experiments, sensitivity analysis of a complicated input-output system is often performed by ranking the so-called Sobol indices. One reason of the popularity of Sobol's approach relies on the simplicity of the statistical estimation of these indices using the so-called Pick and Freeze method. In this work we propose and study sensitivity indices for the case where the output lies on a Riemannian manifold. These indices are based on a Cram\'er von Mises like criterion that takes into account the geometry of the output support. We propose a Pick-Freeze like estimator of these indices based on an $U$--statistic. The asymptotic properties of these estimators are studied. Further, we provide and discuss some interesting numerical examples.

中文翻译：

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黎曼流形输出的灵敏度指数

在计算机代码实验的背景下，复杂的输入输出系统的敏感性分析通常通过对所谓的 Sobol 指数进行排序来执行。Sobol 方法流行的原因之一是使用所谓的 Pick and Freeze 方法对这些指数进行统计估计的简单性。在这项工作中，我们针对输出位于黎曼流形的情况提出并研究了敏感性指数。这些指数基于 Cram\'er von Mises 之类的标准，该标准考虑了输出支持的几何形状。我们基于 $U$--statistic 提出了一个类似 Pick-Freeze 的这些指数的估计器。研究了这些估计量的渐近特性。此外，我们提供并讨论了一些有趣的数值例子。