Studentising Kendall's Tau: U-Statistic Estimators and Bias Correction for a Generalised Rank Variance-Covariance framework

TL;DR

This paper develops a bias-corrected U-statistic estimator for Kemeny's rank correlation, addressing ties and providing more accurate inference in ordinal data analysis.

Contribution

It introduces a complete U-statistic estimator for Kemeny's au with ties and a consistent standard error estimator, improving statistical properties over existing methods.

Findings

Estimator reduces bias in tied data scenarios

Null distribution follows a t-distribution with N-2 degrees of freedom

Outperforms Kendall's au_b in simulation studies

Abstract

Kemeny (1959) introduced a topologically complete metric space to study ordinal random variables, particularly in the context of Condorcet's paradox and the measurability of ties. Building on this, Emond & Mason (2002) reformulated Kemeny's framework into a rank correlation coefficient by embedding the metric space into a Hilbert structure. This transformation enables the analysis of data under weak order-preserving transformations (monotonically non-decreasing) within a linear probabilistic framework. However, the statistical properties of this rank correlation estimator, such as bias, estimation variance, and Type I error rates, have not been thoroughly evaluated. In this paper, we derive and prove a complete U-statistic estimator in the presence of ties for Kemeny's \(\tau_{\kappa}\), addressing the positive bias introduced by tied ranks. We also introduce a consistent population standard error estimator. The null distribution of the test statistic is shown to follow a \(t_{(N-2)}\)-distribution. Simulation results demonstrate that the proposed method outperforms Kendall's \(\tau_{b}\), offering a more accurate and robust measure of ordinal association which is topologically complete upon standard linear models.