On copula density estimation and measures of multivariate association

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Measuring the association between random variables is crucial in risk management and financial modeling. Traditional measures like Spearman's rho and Kendall's tau depend solely on the copula, independent of univariate marginal distributions. In contrast, mutual information, rooted in information theory, offers a distinct approach to measuring association. While it also remains unaffected by univariate margins, mutual information is based on the copula density rather than the copula itself. This work explores the theoretical properties of mutual information as a measure of multivariate association and examines methods to estimate the copula density from observations of continuous distributions. To address boundary bias, new estimators are introduced, and existing functionals are generalized for multivariate contexts. The performance of these estimators is assessed against standard kernel density estimation methods. Additionally, an algorithm is proposed to enhance variance estimation through resampling techniques like bootstrapping, significantly improving computation time. Often, analysts encounter incomplete continuous data, leading to the investigation of copula and density estimation from contingency tables. The newly developed estimators are applied to estimate Spearman's rho and Kendall's tau, with a comparison of their performance.