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 $\alpha \in (0,1)$. 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.
翻译:基于复制内核希尔伯特空间(又称Hilbert-Schmidt 独立标准)和代号HSIC的自足措施,在统计上被广泛使用,用来从统计上决定两个随机矢量是否依赖。最近,对独立进行了非对称的HSIC的统计测试。然而,这些测试导致与HSIC相关的内核的选择问题。特别是,目前还没有方法客观地选择带有理论保证的具体内核,包括第一和第二类错误。这项工作的主要贡献之一是开发基于HSIC的新的基于HSIC的综合程序,避免这种内核选择,并为这一程序提供理论上的保证。为了实现这一点,我们首先采用基于高西亚内核内核的不防疫单项测试,而该内核的硬核内核含量为0.1美元。从理论上看,我们在SOBlev和Nikolskii 的双向双向双向双向双向双向内核,然后我们进行一系列的自上值和上等的直线性内部测试。随后,我们进行一些定期的直线性、最后的单项测试,然后进行一些直线性、直线测试,从而的直线测试,然后进行比级的直线的直线性测试,并进行比的直等的直线性测试。