Studentising Kendall's Tau: U-Statistic Estimators and Bias Correction for a Generalised Rank Variance-Covariance framework
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917174472343552 |
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| author | Hurley, Landon |
| author_facet | Hurley, Landon |
| contents | 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 \(τ_κ\), 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 \(τ_{b}\), offering a more accurate and robust measure of ordinal association which is topologically complete upon standard linear models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_10973 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Studentising Kendall's Tau: U-Statistic Estimators and Bias Correction for a Generalised Rank Variance-Covariance framework Hurley, Landon Methodology Statistics Theory 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 \(τ_κ\), 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 \(τ_{b}\), offering a more accurate and robust measure of ordinal association which is topologically complete upon standard linear models. |
| title | Studentising Kendall's Tau: U-Statistic Estimators and Bias Correction for a Generalised Rank Variance-Covariance framework |
| topic | Methodology Statistics Theory |
| url | https://arxiv.org/abs/2307.10973 |