Adaptive test of independence based on HSIC measures

Abstract

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 the choice of the kernels associated to the HSIC. In particular, there is as yet no method to objectively select specific kernels with theoretical guarantees in terms of first and second kind errors. One of the main contributions of this work is to develop a new HSIC-based aggregated procedure which avoids such a kernel choice, and to provide theoretical guarantees for this procedure. To achieve this, we first introduce non-asymptotic single tests based on Gaussian kernels with a given bandwidth, which are of prescribed level α ∈ (0, 1). From a theoretical point of view, we upper-bound their uniform separation rate of testing over Sobolev and Nikol'skii balls. Then, we aggregate several single tests, and obtain similar upper-bounds for the uniform separation rate of the aggregated procedure over the same regularity spaces. Another main contribution is that we provide a lower-bound for the non-asymptotic minimax separation rate of testing over Sobolev balls, and deduce that the aggregated procedure is adaptive in the minimax sense over such regularity spaces. Finally, from a practical point of view, we perform numerical studies in order to assess the efficiency of our aggregated procedure and compare it to existing independence tests in the literature.

Discussion of 'Multivariate Fisher's independence test for multivariate dependence'

“…Studying power is thus necessarily more subjective than studying validity, but once we have formulated our assumptions and priorities it is possible to search for tests with optimal power. The most common approaches to nonparametric statistics impose smoothness conditions, for example Sobolev smoothness, and prioritize alternative distributions that are both smooth and exhibit relatively strong dependence Albert et al, 2022). However, there are other choices: Zhang (2019) imposes conditions to assume that alternatives exhibit dependence early on in the binary expansions of the data, and the current work assumes, for example in Theorem 3, that alternatives exhibit dependence at relatively coarse resolutions.…”

A general Monte Carlo method for multivariate goodness-of-fit testing applied to elliptical families

“…For this issue we refer to Henze (1997), Henze et al (2019), Tenreiro (2009Tenreiro ( , 2019, and Allison and Santana (2015). Now the more general issue of the proper choice of the specific functional form of the weight function has been investigated to some degree by Lindsay et al (2014) and Albert et al (2022), but the results are somewhat far from having an immediate impact on actual test implementation. We close on the note that although the emphasis herein is for elliptical families, the new tests may be applied to alternative testing situations (say with discrete distributions), and in more general settings, such as for instance in families of distributions over manifolds.…”

Optimized HSIC‐based tests for sensitivity analysis: Application to thermalhydraulic simulation of accidental scenario on nuclear reactor

“…Finally, the third criterion consists in checking the two previous ones in order to guarantee a good ordering and a good screening. The algorithm is similar to Algorithm 3 but with the following stopping criterion (step 10 of Algorithm 3): with 𝐫 𝐵 and 𝑒 𝐵 𝑖 defined by Equation (7) and Equation ( 5), respectively. It should be noted that this criterion, more demanding to be satisfied, may require more permutations.…”