On the epsilon-delta Structure Underlying Chatterjee's Rank Correlation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912762241744896 |
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| author | Sato, Zeusu |
| author_facet | Sato, Zeusu |
| contents | We provide an epsilon-delta interpretation of Chatterjee's rank correlation by tracing its origin to a notion of local dependence between random variables. Starting from a primitive epsilon-delta construction, we show that rank-based dependence measures arise naturally as epsilon to zero limits of local averaging procedures. Within this framework, Chatterjee's rank correlation admits a transparent interpretation as an empirical realization of a local L1 residual.
We emphasize that the probability integral transform plays no structural role in the underlying epsilon-delta mechanism, and is introduced only as a normalization step that renders the final expression distribution-free. We further consider a moment-based analogue obtained by replacing the absolute deviation with a squared residual. This L2 formulation is independent of rank transformations and, under a Gaussian assumption, recovers Pearson's coefficient of determination. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12363 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the epsilon-delta Structure Underlying Chatterjee's Rank Correlation Sato, Zeusu Statistics Theory Probability Methodology 62H20, 60E05 We provide an epsilon-delta interpretation of Chatterjee's rank correlation by tracing its origin to a notion of local dependence between random variables. Starting from a primitive epsilon-delta construction, we show that rank-based dependence measures arise naturally as epsilon to zero limits of local averaging procedures. Within this framework, Chatterjee's rank correlation admits a transparent interpretation as an empirical realization of a local L1 residual. We emphasize that the probability integral transform plays no structural role in the underlying epsilon-delta mechanism, and is introduced only as a normalization step that renders the final expression distribution-free. We further consider a moment-based analogue obtained by replacing the absolute deviation with a squared residual. This L2 formulation is independent of rank transformations and, under a Gaussian assumption, recovers Pearson's coefficient of determination. |
| title | On the epsilon-delta Structure Underlying Chatterjee's Rank Correlation |
| topic | Statistics Theory Probability Methodology 62H20, 60E05 |
| url | https://arxiv.org/abs/2512.12363 |