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We propose a novel statistical test to assess the mutual independence ofmultidimensional random vectors. Our approach is based on the $L_1$-distancebetween the joint density function and the product of the marginal densitiesassociated with the presumed independent vectors. Under the null hypothesis, weemploy Poissonization techniques to establish the asymptotic normalapproximation of the corresponding test statistic, without imposing anyregularity 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 ourmethod. Moreover, our tests exhibit nontrivial local power against a subset oflocal alternatives, which converge to the null hypothesis at a rate of {${\tinyn^{\tiny -1/2}h_n^{\tiny -{d/4}}}$}, $d\geq 2$, where $n$ represents the samplesize and $h_n$ denotes the bandwidth. Finally, the theory is supported by acomprehensive simulation study to investigate the finite-sample performance ofour proposed test. The results demonstrate that our testing procedure generallyoutperforms existing approaches across various examined scenarios.

A nonparametric distribution-free test of independence among continuous random vectors based on \texorpdfstring{$L_1$}{}-norm