Asymptotic representations for Spearman's footrule correlation coefficient

Liqi Xia; Li Guan; Weimin Xu; Liqi Xia; Li Guan; Weimin Xu

doi:10.3934/era.2025220

Electronic Research Archive

2025, Volume 33, Issue 8: 4893-4916. doi: 10.3934/era.2025220

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Research article

Asymptotic representations for Spearman's footrule correlation coefficient

Liqi Xia ¹,
Li Guan ²,
Weimin Xu ^{3
,
,}

1.
School of Mathematics, Qilu Normal University, Jinan 250200, China
2.
School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing 100124, China
3.
School of Data Science and Artificial Intelligence, Wenzhou University of Technology, Wenzhou 325000, China

Received: 27 June 2025 Revised: 04 August 2025 Accepted: 08 August 2025 Published: 21 August 2025

This study addressed the theoretical challenges faced by rank-dependence structures in Spearman's footrule correlation coefficient. Two asymptotic representations were proposed to approximate its null distribution. The first approach simplifies dependencies by substituting empirical distribution functions with their population counterparts. The second employs Hájek projection technique to decompose the initial form into a sum of independent components, establishing rigorous asymptotic normality. Simulation experiments combined with real-world data analyses validated both representations, demonstrating their excellent approximation to the limiting normal distribution under the independence hypothesis.
- asymptotic representation,
- Spearman's footrule,
- rank correlation,
- correlation coefficient,
- Hájek projection
Citation: Liqi Xia, Li Guan, Weimin Xu. Asymptotic representations for Spearman's footrule correlation coefficient[J]. Electronic Research Archive, 2025, 33(8): 4893-4916. doi: 10.3934/era.2025220

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Abstract

This study addressed the theoretical challenges faced by rank-dependence structures in Spearman's footrule correlation coefficient. Two asymptotic representations were proposed to approximate its null distribution. The first approach simplifies dependencies by substituting empirical distribution functions with their population counterparts. The second employs Hájek projection technique to decompose the initial form into a sum of independent components, establishing rigorous asymptotic normality. Simulation experiments combined with real-world data analyses validated both representations, demonstrating their excellent approximation to the limiting normal distribution under the independence hypothesis.

References

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