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

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.