{"ID":2831646,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.07443","arxiv_id":"2512.07443","title":"A multivariate extension of Azadkia-Chatterjee's rank coefficient","abstract":"The Azadkia-Chatterjee coefficient is a rank-based measure of dependence between a random variable $Y \\in \\mathbb{R}$ and a random vector ${\\boldsymbol Z} \\in \\mathbb{R}^{d_Z}$. In this paper, we propose a multivariate extension that measures the dependence between random vectors ${\\boldsymbol Y} \\in \\mathbb{R}^{d_Y}$ and ${\\boldsymbol Z} \\in \\mathbb{R}^{d_Z}$, based on $n$ i.i.d. samples. The proposed coefficient converges almost surely to a limit with the following properties: i) it lies in $[0, 1]$; ii) it is equal to zero if and only if ${\\boldsymbol Y}$ and ${\\boldsymbol Z}$ are independent; and iii) it is equal to one if and only if ${\\boldsymbol Y}$ is almost surely a function of ${\\boldsymbol Z}$. Remarkably, the only assumption required by this convergence is that ${\\boldsymbol Y}$ is not almost surely a constant vector. We further prove that under the same mild condition and after a proper scaling, this coefficient converges in distribution to a standard normal random variable when ${\\boldsymbol Y}$ and ${\\boldsymbol Z}$ are independent. This asymptotic normality result allows us to construct a Wald-type hypothesis test of independence based on this coefficient. To compute this coefficient, we propose a merge sort based algorithm that runs in $O(n (\\log n)^{d_Y})$. Finally, we show that it can be used to measure the conditional dependence between ${\\boldsymbol Y}$ and ${\\boldsymbol Z}$ conditional on a third random vector ${\\boldsymbol X}$, and prove that the measure is monotonic with respect to the deviation from an independence distribution under certain model restrictions.","short_abstract":"The Azadkia-Chatterjee coefficient is a rank-based measure of dependence between a random variable $Y \\in \\mathbb{R}$ and a random vector ${\\boldsymbol Z} \\in \\mathbb{R}^{d_Z}$. In this paper, we propose a multivariate extension that measures the dependence between random vectors ${\\boldsymbol Y} \\in \\mathbb{R}^{d_Y}$...","url_abs":"https://arxiv.org/abs/2512.07443","url_pdf":"https://arxiv.org/pdf/2512.07443v2","authors":"[\"Wenjie Huang\",\"Zonghan Li\",\"Yuhao Wang\"]","published":"2025-12-08T11:17:18Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"stat.ME\"]","methods":"[]","has_code":false}