条件独立性并非（完全）逐点可检验

中文摘要 AI 辅助

Shah 和 Peters 证明了，具有有限样本或一致控制水平的条件独立性检验，对任何备择假设只有平凡的功效。然而，大多数实际检验仅声称具有逐点渐近水平。文献中曾出现关于具有逐点渐近水平且对任何备择假设具有一致性的条件独立性检验的错误断言；此类检验是否真正存在一直悬而未决。我们解决了这个问题。即使将假设限制在欧几里得空间紧子集上具有有界密度的分布，对于任何具有逐点渐近水平 $\alpha$ 的（可能随机的）检验序列，对任意 $\varepsilon > 0$，都存在一个条件相依分布，使得该检验的 limsup 功效至多为 $\alpha + \varepsilon$。当水平控制仅要求对具有连续密度和一致连续条件分布的条件独立分布成立，且条件相依分布具有光滑密度时，上述结论仍然成立。另一方面，一致水平检验对任何备择假设只有平凡功效这一极端限制并不适用于逐点水平检验。我们展示了一个检验，在无正则性假设下，具有逐点渐近水平 $\alpha$，对所有备择假设具有严格更高的功效，并且对所有相依程度超过所选标量阈值的分布，功效趋于 1。

英文摘要

Shah and Peters showed that a conditional independence test with finite-sample or uniformly-controlled level has only trivial power against any alternative. Most practical tests, however, only claim pointwise asymptotic level. There have been incorrect claims in the literature of conditional independence tests with pointwise asymptotic level and consistency against any alternative; whether such a test actually exists has remained open. We resolve this question. Even restricting to hypotheses with a bounded density on compact subsets of Euclidean spaces, for any sequence of (possibly randomized) tests with pointwise asymptotic level $α$, for every $\varepsilon > 0$ there is a conditionally dependent distribution where the test's limsup power is at most $α+ \varepsilon$. This remains true when the level control is required only over conditionally independent distributions with a continuous density and a uniformly continuous conditional law, and the conditionally dependent distributions have smooth densities. On the other hand, the extreme limitation of trivial power against any alternative for uniform-level tests does not apply to tests with pointwise level. We exhibit a test that, without regularity assumptions, has pointwise asymptotic level $α$, strictly higher power for all alternatives, and power tending to one for all distributions whose dependence exceeds a chosen scalar threshold.

发表机构

• University of British Columbia（不列颠哥伦比亚大学）

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