Conditional independence (CI) testing arises naturally in many scientific problems and applications domains. The goal of this problem is to investigate the conditional independence between a response variable $Y$ and another variable $X$, while controlling for the effect of a high-dimensional confounding variable $Z$. In this paper, we introduce a novel test, called `Pearson Chi-squared Conditional Randomization' (PCR) test, which uses the distributional information on covariates $X,Z$ and constructs randomizations to test conditional independence. PCR leverages the i.i.d-ness property of the observations to obtain high-resolution p-values with a very small number of conditional randomizations. We also provide a power analysis of the PCR test, which captures the effect of various parameters of the test, the sample size and the distance of the alternative from the set of null distributions, measured in terms of a notion called `conditional relative density'. In addition, we propose two extensions of the PCR test, with important practical implications: $(i)$ parameter-free PCR, which uses Bonferroni's correction to decide on a tuning parameter in the test; $(ii)$ robust PCR, which avoids inflations in the size of the test when there is slight error in estimating the conditional law $P_{X|Z}$.

翻译：条件独立性检验自然出现在许多科学问题和应用领域中。该问题的目标是探究响应变量$Y$与另一变量$X$之间的条件独立性，同时控制高维混杂变量$Z$的影响。本文提出了一种称为“皮尔逊卡方条件随机化”检验的新方法，该方法利用协变量$X,Z$的分布信息，通过构建随机化过程来检验条件独立性。PCR利用观测值的独立同分布特性，以极少次数的条件随机化获得高分辨率p值。我们还对PCR检验进行了功效分析，该分析捕捉了检验的各项参数、样本量以及备择分布与零分布集合之间距离（通过一种称为“条件相对密度”的概念度量）的影响。此外，我们提出了PCR检验的两个具有重要实际意义的扩展：$(i)$无参数PCR，其使用邦费罗尼校正来确定检验中的调优参数；$(ii)$稳健PCR，其在估计条件律$P_{X|Z}$存在微小误差时可避免检验规模的膨胀。