Abstract

Quantifying the strength of functional dependence between random scalars X and Y is a fundamental statistical problem. While many existing correlation coefficients effectively identify linear or monotone functional dependence, they often fail to capture more general, non-monotonic dependencies. In response, we propose a family of simple and computationally efficient correlation coefficients that encompasses a wide class of novel dependency measures, including the Chatterjee’s correlation coefficient as a special case. We prove that each of our correlation coefficients converges to a population-level correlation measure as the sample size grows, and a central limit theorem is established under the assumption of independence between X and Y. Moreover, we provide the necessary and sufficient condition for the correlation measure to satisfy three desirable properties: (P1). it belongs to the range of [0, 1]; (P2). it equals 1 if and only if Y is a measurable function of X; and (P3). it equals 0 if and only if Y is independent of X. Based on both theoretical analysis and empirical evidence, our correlation coefficients demonstrate superior practicality, robustness, and computational efficiency compared with others.

Key words and phrases: Central limit theorem, correlation coefficient, functional dependence, independence, rank-based correlation

Information

Preprint No.SS-2026-0108
Manuscript IDSS-2026-0108
Complete AuthorsMuhong Gao, Qizhai Li
Corresponding AuthorsQizhai Li
Emailsliqz@amss.ac.cn

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Acknowledgments

We thank the associate editor and the two reviewers for their valuable comments and suggestions. Muhong Gao’s work was supported by National Natural Science Foundation of China [Grant No. 12601499], and a startup fund from University of International Business and Economics. Qizhai Li’s work was supported by National Natural Science Foundation of China [Grant No. 12325110, 12288201], the robotic AI-Scientist platform of Chinese Academy of Sciences. Qizhai Li is also affiliated with the School of Mathematical Sciences, University of Chinese Academy of Sciences.

Supplementary Materials

The Supplementary Material contains the proofs of main results.

Supplementary materials are available for download.