In this paper, the defining properties of any valid measure of the dependence between two continuous random variables are revisited and complemented with two original ones, shown to imply other usual postulates. While other popular choices are proved to violate some of these requirements, a class of dependence measures satisfying all of them is identified. One particular measure, that we call the Hellinger correlation, appears as a natural choice within that class due to both its theoretical and intuitive appeal. A simple and efficient nonparametric estimator for that quantity is proposed, with its implementation publicly available in the R package HellCor. Synthetic and real-data examples illustrate the descriptive ability of the measure, which can also be used as test statistic for exact independence testing.

The concept of depth has proved very important for multivariate and functional data analysis, as it essentially acts as a surrogate for the notion of ranking of observations which is absent in more than one dimension. Motivated by the rapid development of technology, in particular the advent of ‘Big Data’, we extend here that concept to general metric spaces, propose a natural depth measure and explore its properties as a statistical depth function. Working in a general metric space allows the depth to be tailored to the data at hand and to the ultimate goal of the analysis, a very desirable property given the polymorphic nature of modern data sets. This flexibility is thoroughly illustrated by several real data analyses.

The kernel estimator is known not to be adequate for estimating the density of a positive random variable X. The main reason is the well-known boundary bias problems that it suffers from, but also its poor behavior in the long right tail that such a density typically exhibits. A natural approach to this problem is to first estimate the density of the logarithm of X, and obtaining an estimate of the density of X using standard results on functions of random variables (“back-transformation”). Although intuitive, the basic application of this idea usually yields poor results, as was documented earlier in the literature. In this article, the main reason for this underachievement is identified, and an easy fix is suggested. It is demonstrated that combining the transformation with local-likelihood density estimation methods produces very good estimators of ${\mathbb{R}}^{+}$-supported densities, not only close to the boundary, but also in the right tail. The asymptotic properties of the proposed “local-likelihood transformation kernel density estimators” are derived for a generic transformation, not only for the logarithm, which allows one to consider other transformations as well. One of them, called the “probex” transformation, is given more focus. Finally, the excellent behavior of those estimators in practice is evidenced through a comprehensive simulation study and the analysis of several real datasets. A nice consequence of articulating the method around local-likelihood estimation is that the resulting density estimates are typically smooth and visually pleasant, without oversmoothing important features of the underlying density.

In this short communication, I share some personal thoughts on Sklar’s theorem and copulas after reading the original paper (Sklar, 1959) in French. After providing a literal translation of Sklar’s original statements, I argue that the modern version of ‘Sklar’s theorem’ given in most references has a slightly different emphasis, which may lead to subtly different interpretations. In particular, with no reference to the subcopula, modern ‘Sklar’s theorem’ does not provide the clues to fully appreciate when the copula representation of a distribution may form a valid basis for dependence modelling and when it may not.