arXiv 1 May 2025 · Statistics — Methodology
arXiv:2505.00785 · PDF · DOI · OpenAlex · Extracted main text
This paper develops an intuitive concept of perfect dependence between two variables of which at least one has a nominal scale that is attainable for all marginal distributions and proposes a set of dependence measures that are 1 if and only if this perfect dependence is satisfied. The advantages of these dependence measures relative to classical dependence measures like contingency coefficients, Goodman-Kruskal's lambda and tau and the so-called uncertainty coefficient are twofold. Firstly, they are defined if one of the variables is real-valued and exhibits continuities. Secondly, they satisfy the property of attainability. That is, they can take all values in the interval [0,1] irrespective of the marginals involved. Both properties are not shared by the classical dependence measures which need two discrete marginal distributions and can in some situations yield values close to 0 even though the dependence is strong or even perfect. Additionally, I provide a consistent estimator for one of the new dependence measures together with its asymptotic distribution under independence as well as in the general case. This allows to construct confidence intervals and an independence test, whose finite sample performance I subsequently examine in a simulation study. Finally, I illustrate the use of the new dependence measure in two applications on the dependence between the variables country and income or country and religion, respectively.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Weiß, Christian H and Göb, Rainer (2008) Measuring serial dependence in categorical time series | 0.843 | 3 | 3 | 100% |
| 2 | Pohle, Marc-Oliver and Wermuth, Jan-Lukas and Weiß, Christian H (2025) Asymptotic Inference for Rank Correlations self | 0.822 | 9 | 4 | 56% |
| 3 | Pohle, Marc-Oliver and Wermuth, Jan-Lukas (2026) Proper Correlation Coefficients for Discrete Random Variables self | 0.811 | 4 | 2 | 100% |
| 4 | Hajo Holzmann and Bernhard Klar (2024) Lancaster correlation – a new dependence measure linked to maximum correlation | 0.737 | 3 | 2 | 100% |
| 5 | Pohle, Marc-Oliver and Dimitriadis, Timo and Wermuth, Jan-Lukas (2024) Measuring Dependence between Events self | 0.737 | 3 | 2 | 100% |
| 6 | Zurlo, Gina A (2025) World Religion Database | 0.644 | 2 | 2 | 100% |
| 7 | Gebelein, Hans (1941) Das statistische Problem der Korrelation als Variations- und Eigenwertproblem und sein Zusammenhang mit der Ausgleichsrechnung | 0.644 | 2 | 2 | 100% |
| 8 | Geenens, Gery (2020) Copula modeling for discrete random vectors | 0.644 | 2 | 2 | 100% |
| 9 | Hirschfeld, Herman Otto (1935) A Connection between Correlation and Contingency | 0.644 | 2 | 2 | 100% |
| 10 | Tchen, André H (1980) Inequalities for distributions with given marginals | 0.644 | 2 | 2 | 100% |
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