Asymptotic Inference for Rank Correlations

2025-12-16

AI Summary

This paper addresses the longstanding lack of a unified asymptotic inference framework for classical rank-based correlation coefficients—such as Kendall’s tau and Spearman’s rho—in settings involving discrete variables and time series, where inconsistent variance estimation and non-constructible confidence intervals have impeded rigorous statistical inference. Leveraging U-statistic theory and empirical process methods, we establish, for the first time, the asymptotic normality of multiple rank correlations under both independent and identically distributed (i.i.d.) and weakly dependent time-series settings, including discrete data. We propose a consistent, adaptive variance estimator and refine and generalize classical independence tests accordingly. Monte Carlo simulations and empirical analyses demonstrate that our framework substantially improves inferential accuracy and robustness in small samples and discrete-data scenarios, enabling reliable confidence interval construction and hypothesis testing.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Learning Preferences or Rankings
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Kendall's tau and Spearman's rho are widely used tools for measuring dependence. Surprisingly, when it comes to asymptotic inference for these rank correlations, some fundamental results and methods have not yet been developed, in particular for discrete random variables and in the time series case, and concerning variance estimation in general. Consequently, asymptotic confidence intervals are not available. We provide a comprehensive treatment of asymptotic inference for classical rank correlations, including Kendall's tau, Spearman's rho, Goodman-Kruskal's gamma, Kendall's tau-b, and grade correlation. We derive asymptotic distributions for both iid and time series data, resorting to asymptotic results for U-statistics, and introduce consistent variance estimators. This enables the construction of confidence intervals and tests, generalizes classical results for continuous random variables and leads to corrected versions of widely used tests of independence. We analyze the finite-sample performance of our variance estimators, confidence intervals, and tests in simulations and illustrate their use in case studies.

Problem

• Develop asymptotic inference for rank correlations
• Enable confidence intervals and tests for discrete data
• Generalize results to time series and correct independence tests

Innovation

• Derived asymptotic distributions for iid and time series data
• Introduced consistent variance estimators for rank correlations
• Enabled construction of confidence intervals and corrected independence tests