2023/07/20 by Landon Hurley, Hurley, Landon
Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Statistics Education and Methodologies #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2307.10973
openalex publication_date 2023/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \(τκ\), 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 \(τb\), offering a more accurate and robust measure of ordinal association which is topologically complete upon standard linear models.