Recently, data depth has been widely used to rank multivariate data. The study of the depth-based $Q$ statistic, originally proposed by Liu and Singh (1993), has become increasingly popular when it can be used as a quality index to differentiate between two samples. Based on the existing theoretical foundations, more and more variants have been developed for increasing power in the two sample test. However, the asymptotic expansion of the $Q$ statistic in the important foundation work of Zuo and He (2006) currently has an optimal rate $m^{-3/4}$ slower than the target $m^{-1}$, leading to limitations in higher-order expansions for developing more powerful tests. We revisit the existing assumptions and add two new plausible assumptions to obtain the target rate by applying a new proof method based on the Hoeffding decomposition and the Cox-Reid expansion. The aim of this paper is to rekindle interest in asymptotic data depth theory, to place Q-statistical inference on a firmer theoretical basis, to show its variants in current research, to open the door to the development of new theories for further variants requiring higher-order expansions, and to explore more of its potential applications.

翻译：近年来，数据深度被广泛用于多元数据的排序研究。基于深度的Q统计量最初由Liu与Singh（1993）提出，当其可作为区分两个样本的质量指标时，相关研究日益受到关注。在现有理论基础之上，学界已发展出越来越多的变体以提升双样本检验的效能。然而，Zuo与He（2006）重要奠基性工作中提出的Q统计量渐近展开，其最优收敛速率目前仅为m^{-3/4}，低于目标速率m^{-1}，这导致在构建更高阶展开以发展更强检验方法时存在局限。本文重新审视现有假设，通过引入两个新的合理假设，并基于Hoeffding分解与Cox-Reid展开的新证明方法，最终获得目标收敛速率。本文旨在重新激发对数据深度渐近理论的研究兴趣，为Q统计推断奠定更坚实的理论基础，展示当前研究中的各类变体，为需要更高阶展开的新变体理论发展开启大门，并进一步探索其潜在应用价值。