AI 中文总结
该研究提出距离剖面嵌入（DPE）方法，构建适配非欧几里得随机对象的独立性与条件独立性检验统一框架，可处理对象值条件变量，避免置换检验高计算负担，经模拟与真实数据验证有效。
AI 中文摘要
独立性或条件独立性检验是统计推断的基础，但现有针对非欧几里得随机对象的方法，常面临几何灵活性与理论可处理性之间的艰难权衡。我们提出距离剖面嵌入（Distance Profile Embedding, DPE），一种将一般度量空间中的随机对象映射到平方可积函数希尔伯特空间的新型表示。我们证明该映射是单射的，且在无需等距希尔伯特嵌入或一一对应条件的情况下保留完整分布信息。利用DPE，我们开发了一个针对随机对象边缘与条件独立性检验的统一框架，该框架对检验水平和功效均具有严格的渐近理论。值得注意的是，我们的框架是文献中首个能在条件独立性检验中适配对象值条件变量的方法，克服了现有方法的欧几里得或希尔伯特约束。我们通过闭式渐近零分布促进解析p值的计算，避免了现有基于度量的方法中常见的置换检验的计算负担。我们的方法的数值性能通过模拟以及两个真实世界应用（分别涉及肠道微生物组组成和全球人类死亡率分布）得到验证。
英文摘要
Testing independence or conditional independence is fundamental to statistical inference, yet existing methods for non-Euclidean random objects often face a difficult trade-off between geometric flexibility and theoretical tractability. We introduce the Distance Profile Embedding (DPE), a novel representation that maps random objects from general metric spaces into a Hilbert space of square-integrable functions. We prove that this mapping is injective and preserves full distributional information without requiring isometric Hilbert embeddings or one-to-one correspondence conditions. Leveraging the DPE, we develop a unified framework for marginal and conditional independence testing of random objects that enjoys a rigorous asymptotic theory for both size and power. Notably, our framework is the first in the literature to accommodate object-valued conditioning variables when testing conditional independence, overcoming the Euclidean or Hilbertian constraints of existing methodologies. We facilitate the calculation of analytic $p$-values using closed-form asymptotic null distributions, which avoids the computational burden of permutation tests common in existing metric-based methods. The numerical properties of our methods are demonstrated through both simulations and two real-world applications involving gut microbiome compositions and global human mortality distributions, respectively.
Comments32 pages, 4 figures
