Aggregated tests of independence based on HSIC measures

International audienceThe presented works are conducted within the framework of a PhD thesis funded by the CEA (French: Commissariat à l'Énergie atomique et aux Énergies alternatives). The objective is to propose new statistical methods for sensitivity analysis in support of safety studies for Sodium-cooled Fast Reactors. This methodology will be applied on a real industrial test case simulating an unprotected primary loss of flow accident.Dependence measures based on reproducing kernel Hilbert spaces, also known as Hilbert-Schmidt Independence Criterion and denoted HSIC, are widely used to statistically decide whether or not two random vectors are dependent. Recently, non-parametric HSIC-based statistical tests of independence have been performed. However, these tests lead to the question of prior choice of the kernels associated to HSIC, since there is as yet no method to objectively select specific kernels. In order to avoid a particular kernel choice, we propose a new HSIC-based aggregated procedure allowing to take into account several Gaussian kernels. To achieve this, we first propose non-asymptotic single tests of level $\alpha \in (0,1)$ and second type error controlled by $\beta \in (0,1)$. We also provide a sharp upper bound of the uniform seperation rate of the proposed tests. Thereafter, we introduce a multiple testing procedure in the case of Gaussian kernels, considering a set of various parameters. These agregated tests are shown to be of level $\alpha$ and to overperform single tests in terms of uniform separation rates