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Asymptotic Inference for Rank Correlations

Marc-Oliver Pohle, Jan-Lukas Wermuth, Christian H. Weiß

TL;DR

This paper tackles the absence of rigorous asymptotic inference for rank-based dependence measures, especially for discrete variables and time series. It leverages U-statistics theory to derive iid and weakly dependent asymptotic distributions for Kendall's tau, Spearman's rho, Goodman-Kruskal's gamma, Kendall's tau_b, and grade correlation, and provides consistent estimators for their asymptotic variances. The authors formulate confidence intervals and hypothesis tests, including simplified variance expressions under independence, and extend results to time-series data via HAC-type long-run variance estimation. Through extensive simulations and two real-data case studies, the work demonstrates robust finite-sample performance and practical applicability, while highlighting remaining challenges in variance estimation under strong dependence and suggesting avenues like self-normalization for future work.

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.

Asymptotic Inference for Rank Correlations

TL;DR

This paper tackles the absence of rigorous asymptotic inference for rank-based dependence measures, especially for discrete variables and time series. It leverages U-statistics theory to derive iid and weakly dependent asymptotic distributions for Kendall's tau, Spearman's rho, Goodman-Kruskal's gamma, Kendall's tau_b, and grade correlation, and provides consistent estimators for their asymptotic variances. The authors formulate confidence intervals and hypothesis tests, including simplified variance expressions under independence, and extend results to time-series data via HAC-type long-run variance estimation. Through extensive simulations and two real-data case studies, the work demonstrates robust finite-sample performance and practical applicability, while highlighting remaining challenges in variance estimation under strong dependence and suggesting avenues like self-normalization for future work.

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.

Paper Structure

This paper contains 35 sections, 18 theorems, 144 equations, 6 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Rank Correlations
  3. Fundamental Concepts: Grades, Sign Differences, and Tie Pro-babilities
  4. Classical Rank Correlations: Kendall's $\tau$ and Spearman's $\rho$
  5. Generalized Rank Correlations for Discrete Random Variables: Goodman-Kruskal's $\gamma$, Kendall's $\tau_b$ and Grade Correlation
  6. Empirical Rank Correlations
  7. Classical Empirical Rank Correlations
  8. Generalized Empirical Rank Correlations for Discrete Random Variables
  9. Asymptotic Theory: IID Data
  10. Consistency
  11. Asymptotic Distributions
  12. Variance Estimation
  13. Confidence Intervals, Tests, and Fisher Transformation
  14. Asymptotic Distributions under Independence
  15. Asymptotic Theory: Time Series
  16. ...and 20 more sections

Key Result

Lemma 2.1

It holds that

Figures (6)

  • Figure 1: Left: Plot of tie probabilities against the mean $\mu$ of a geometric distribution. Right: Plot of asymptotic variances from Corollaries \ref{['cor:iid_and_independent_processes_asymptotic_distribution_tau']} and \ref{['cor:iid_and_independent_processes_asymptotic_distribution_gamma']} when $X$ and $Y$ are independent and identically geometrically distributed. The horizontal lines at 1 and 4/9 correspond to the variances of $\rho_b$ and $\tau_{b,\textup{mod}}$, respectively.
  • Figure 2: Bubble plot visualizing the ecological data example: The x-axis represents the number of plants of the species Lacistema aggregatum and the y-axis the number of plants of the species Protium guianense in each of 100 contiguous quadrants. The bubbles contain and their sizes visualize the counts of certain combinations of x- and y-values.
  • Figure 3: Bubble plot: The x-axis represents the number of daytime and the y-axis the number of nighttime road accidents in the Schiphol area (Netherlands) during the year 2001. The bubbles contain and their sizes visualize the counts of certain combinations of x- and y-values.
  • Figure 4: Time series plot: The x-axis measures the days in 2001 and the y-axis the numbers of road accidents that occurred during that day in the Schiphol area (Netherlands) either during daytime or during nighttime.
  • Figure D.1: Rejection rates for independence tests based on Kendall's $\tau$, Pearson's $r$ and Spearman's $\rho$ for DGPs (\ref{['eq:normaliidDGP']}), (\ref{['eq:t1iidDGP']}) and (\ref{['eq:Zipf1iidDGP']}) with $MC=1,000$ simulation runs.
  • ...and 1 more figures

Theorems & Definitions (42)

  • Lemma 2.1
  • Definition 2.2: Kendall's $\tau$
  • Definition 2.3: Spearman's $\rho$
  • Definition 2.4: Goodman-Kruskal's $\gamma$
  • Definition 2.5: Kendall's $\tau_b$
  • Definition 2.6: Grade Correlation
  • Definition 3.1: Empirical Kendall's $\tau$
  • Definition 3.2: Empirical Spearman's $\rho$
  • Definition 3.3: Empirical Goodman-Kruskal's $\gamma$
  • Definition 3.4: Empirical Kendall's $\tau_b$
  • ...and 32 more