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.

翻译：Kendall's tau 与 Spearman's rho 是衡量相关性的常用工具。令人惊讶的是，关于这些秩相关性的渐近推断，一些基本结果与方法尚未建立，特别是针对离散随机变量与时间序列情形，以及一般性的方差估计问题。因此，渐近置信区间尚不可得。本文对经典秩相关性（包括 Kendall's tau、Spearman's rho、Goodman-Kruskal's gamma、Kendall's tau-b 以及等级相关）的渐近推断进行了全面研究。我们基于 U-统计量的渐近理论，推导了独立同分布数据与时间序列数据的渐近分布，并引入了一致的方差估计量。这使得置信区间的构建与假设检验成为可能，推广了连续随机变量的经典结果，并导出了广泛使用的独立性检验的修正版本。我们通过模拟分析了方差估计量、置信区间及检验方法的有限样本性能，并通过案例研究展示了其应用。