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June 2020 The hardness of conditional independence testing and the generalised covariance measure
Rajen D. Shah, Jonas Peters
Ann. Statist. 48(3): 1514-1538 (June 2020). DOI: 10.1214/19-AOS1857

Abstract

It is a common saying that testing for conditional independence, that is, testing whether whether two random vectors $X$ and $Y$ are independent, given $Z$, is a hard statistical problem if $Z$ is a continuous random variable (or vector). In this paper, we prove that conditional independence is indeed a particularly difficult hypothesis to test for. Valid statistical tests are required to have a size that is smaller than a pre-defined significance level, and different tests usually have power against a different class of alternatives. We prove that a valid test for conditional independence does not have power against any alternative.

Given the nonexistence of a uniformly valid conditional independence test, we argue that tests must be designed so their suitability for a particular problem may be judged easily. To address this need, we propose in the case where $X$ and $Y$ are univariate to nonlinearly regress $X$ on $Z$, and $Y$ on $Z$ and then compute a test statistic based on the sample covariance between the residuals, which we call the generalised covariance measure (GCM). We prove that validity of this form of test relies almost entirely on the weak requirement that the regression procedures are able to estimate the conditional means $X$ given $Z$, and $Y$ given $Z$, at a slow rate. We extend the methodology to handle settings where $X$ and $Y$ may be multivariate or even high dimensional. While our general procedure can be tailored to the setting at hand by combining it with any regression technique, we develop the theoretical guarantees for kernel ridge regression. A simulation study shows that the test based on GCM is competitive with state of the art conditional independence tests. Code is available as the R package $\mathtt{GeneralisedCovarianceMeasure}$ on CRAN.

Citation

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Rajen D. Shah. Jonas Peters. "The hardness of conditional independence testing and the generalised covariance measure." Ann. Statist. 48 (3) 1514 - 1538, June 2020. https://doi.org/10.1214/19-AOS1857

Information

Received: 1 April 2018; Revised: 1 April 2019; Published: June 2020
First available in Project Euclid: 17 July 2020

zbMATH: 07241601
MathSciNet: MR4124333
Digital Object Identifier: 10.1214/19-AOS1857

Subjects:
Primary: 62G10
Secondary: 62G08

Keywords: Conditional independence , Hypothesis testing , kernel ridge regression , testability , wild bootstrap

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 3 • June 2020
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