Conference Agenda

We propose a margin-free measure of bivariate association generalizing Spearman’s rho to the case of non- monotonic dependence that is defined in terms of two square integrable functions on the unit interval. We investigate properties of generalized Spearman correlation when the functions are piecewise continuous and strictly monotonic, with particular focus on the special cases where the functions are drawn from orthonormal bases defined by Legendre polynomials and cosine functions. For continuous random variables, generalized Spearman correlation is treated as a copula-based measure and shown to depend on a pair of uniform-distribution-preserving (udp) transformations determined by the underlying functions. We derive bounds for generalized Spearman correlation and we use a novel technique that we refer to as stochastic inversion of udp transformations to construct singular copulas that attain the bounds and parametric copulas with densities that interpolate between the bounds and model different degrees of non-monotonic dependence.

We also propose sample analogues of generalized Spearman correlation and investigate their asymptotic and small-sample properties. Potential applications of the theory are demonstrated including: exploratory analyses of the dependence structures of datasets and their symmetries; elicitation of functions maximizing generalized Spearman correlation via expansions in orthonormal basis functions; and construction of tractable probability densities to model a wide variety of non-monotonic dependencies.

Extending bivariate dependence concepts to higher dimensions is a challenging but essential task for a comprehensive understanding of multivariate dependence. Moreover, measuring overall dependence based on averages across the full domain of the joint distribution may fail to discern changes in dependence across different segments of the distribution, especially in the tails. In order to incorporate these features, we present the multivariate tail concentration function (TCF) as a graphical tool to assess both global and tail dependence. We show that this tool allows to represent multivariate dependence in a 2D plot regardless of the number of dimensions, it quantifies both lower and upper tail dependence at a finite scale, and it relates to multivariate Blomqvist’s beta. We propose to estimate the TCF non-parametrically using two methods and we compare their finite sample performance through a simulation study. To illustrate its practical application, we use the TCF to evaluate co-movements among the six Spanish banks included in the IBEX35 stock index.

In multivariate regression analysis, the multiple linear correlation coefficient is a commonly used association measure. This measure focuses on a linear relationship between a response variable and predictor variables. When moving away from the linearity of the functional relationship, then we arrive at Kendall's tau and multivariate versions, among others. In an earlier paper by the author (2021), the Kendall regression coefficient was introduced. Here, we extend the coefficient to vector responses Y and discuss properties of it. The coefficient we introduce describes to what degree the response variable Y can be approximated by a monotonous function of the regressors. These regressors are combined in a random vector. One advantage of this approach is that the association measure is based only on the copula (does not depend on marginal distributions), and is hence robust against outliers.
In the talk, we provide a series of interesting properties of the new Kendall coefficient, which are based on those described in Schmid et al. (2010). It turns out that it makes sense to consider Kendall coefficients w.r.t. the various directions of the regressors. Furthermore, we introduce an estimator for the Kendall coefficient, for which a central limit theorem is provided. It is demonstrated that the multivariate extension of Kendall's tau coefficients can be employed in multiple non-linear regression analysis and in dependence modelling, even in the situation of higher dimensions.
Literature:
Schmid, F.; Schmidt, R.; Blumentritt, T.; Gaißer, S.; Ruppert, M. (2010). Copula-based measures of multivariate association. in F. Durante, W. Härdle, P. Jaworski, T. Rychlik (eds.) Copula Theory and Its Applications. Springer Berlin, 2010.
Liebscher, E. (2021). Kendall regression coefficient. Computational Statistics and Data Analysis 157. 107140