We study projection-based diagnostics for distinguishing directional asymmetry from tail-ratio departure in multivariate data. The procedure reduces the problem to one-dimensional projections and computes two quantile-based summaries: a directional skewness measure evaluated over several quantile levels, and an interquantile tail-ratio evaluated relative to a chosen benchmark. The two summaries lead to a four-regime classification: symmetric benchmark-tail, symmetric tail-departed, skewed benchmark-tail, and skewed tail-departed. The quantile formulation avoids relying on third and fourth moments, which can be unstable in heavy-tailed settings. We establish population properties under central symmetry and ellipticity, uniform finite-sample bounds over the searched directions, and consistency of the threshold classifier under separated regimes. A sparse rank-one calculation is also used to show why coordinate directions can complement random directions in high dimensions. The resulting diagnostic is meant to guide subsequent modelling choices, for example whether a symmetric, skewed, tail-departed, or combined multivariate model is appropriate.

翻译：我们研究了基于投影的多元数据方向不对称性与尾部比率偏离的区分诊断方法。该流程将问题简化为单维投影，并计算两个分位数摘要指标：覆盖多个分位数水平的方向偏度度量，以及相对于选定基准的跨分位数尾部比率。这两个摘要指标可导出四类划分：对称基准尾部、对称尾部偏离、偏斜基准尾部和偏斜尾部偏离。该分位数公式避免依赖三阶与四阶矩（在重尾情形下可能不稳定）。我们建立了中心对称与椭圆分布下的总体性质、搜索方向上的均匀有限样本界，以及分离区域下阈值分类器的一致性。通过稀疏秩一计算进一步阐释了高维情形下坐标方向为何可补充随机方向。该诊断方法旨在指导后续建模选择，例如判断对称、偏斜、尾部偏离或组合多元模型是否适用。