We propose a novel statistical test to assess the mutual independence of multidimensional random vectors. Our approach is based on the $L_1$-distance between the joint density function and the product of the marginal densities associated with the presumed independent vectors. Under the null hypothesis, we employ Poissonization techniques to establish the asymptotic normal approximation of the corresponding test statistic, without imposing any regularity assumptions on the underlying Lebesgue density function, denoted as $f(\cdot)$. Remarkably, we observe that the limiting distribution of the $L_1$-based statistics remains unaffected by the specific form of $f(\cdot)$. This unexpected outcome contributes to the robustness and versatility of our method. Moreover, our tests exhibit nontrivial local power against a subset of local alternatives, which converge to the null hypothesis at a rate of {${\tiny n^{\tiny -1/2}h_n^{\tiny -{d/4}}}$}, $d\geq 2$, where $n$ represents the sample size and $h_n$ denotes the bandwidth. Finally, the theory is supported by a comprehensive simulation study to investigate the finite-sample performance of our proposed test. The results demonstrate that our testing procedure generally outperforms existing approaches across various examined scenarios.

翻译：我们提出一种新颖的统计检验方法，用于评估多维随机向量的相互独立性。该方法基于联合密度函数与假定独立向量对应的边际密度乘积之间的$L_1$距离。在原假设下，我们采用泊松化技术建立相应检验统计量的渐近正态近似，且无需对底层的勒贝格密度函数$f(\cdot)$施加任何正则性假设。值得注意的是，我们观察到基于$L_1$的统计量的极限分布不受$f(\cdot)$具体形式的影响。这一意外发现增强了我们方法的稳健性和通用性。此外，我们的检验对以速率${\tiny n^{\tiny -1/2}h_n^{\tiny -{d/4}}}$（$d\geq 2$，其中$n$为样本容量，$h_n$为带宽）收敛于原假设的一类局部备择假设子集具有非平凡局部功效。最后，我们通过全面的模拟研究验证该理论，考察所提检验的有限样本性能。结果表明，在多种测试场景下，我们的检验方法普遍优于现有方法。