  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.
