Conference Agenda

Kendall's Tau and Spearman's Rho are key tools for measuring dependence between two variables. Surprisingly, when it comes to statistical inference for these rank correlations, some fundamental results and methods have not been developed, in particular regarding asymptotic variances for discrete random variables or in the time series case, and variance estimation in general such that e.g. asymptotic confidence intervals are not available. We provide a comprehensive treatment of asymptotic inference for classical rank correlations, including Kendall's Tau, Spearman's Rho, Goodman and Kruskal's Gamma, Kendall's Tau-b and Grade Correlation. We derive asymptotic distributions and variances for both independent and time-series data, resorting to asymptotic results for U-statistics, and introduce consistent variance estimators. This enables the construction of confidence intervals and tests. We also obtain limiting variances under independence between the two variables or processes of interest, which generalize classical results limited to continuous random variables and lead to corrected versions of widely-used tests of independence based on rank correlations. We analyze the finite-sample performance of our variance estimators, confidence intervals and tests in simulations and illustrate their use in applications.

Recently established, directed dependence measures for pairs $(X,Y)$ of random variables build upon the natural idea of comparing the conditional distributions of $Y$ given $X=x$ with the unconditional distribution of $Y$. They assign pairs $(X,Y)$ values in $[0,1]$, where the value is $0$ if and only if $X,Y$ are independent, and it is $1$ exclusively for $Y$ being a measurable function of $X$. Here we show that comparing randomly drawn conditional distributions with each other instead or, equivalently, analyzing how sensitive the conditional distribution of $Y$ given $X=x$ is on $x$, opens the door to constructing novel families of dependence measures $\Lambda_\varphi$ induced by general convex functions $\varphi: \mathbb{R} \rightarrow \mathbb{R}$, containing, e.g., Chatterjee's coefficient of correlation as special case. After establishing additional useful properties of $\Lambda_\varphi$ we focus on continuous $(X,Y)$, translate $\Lambda_\varphi$ to the copula setting, consider the $L^p$-version and establish an estimator which is strongly consistent in full generality. A real data example and a simulation study illustrate the chosen approach and the performance of the estimator. Complementing the afore-mentioned results, we show how a slight modification of the sensitivity idea underlying $\Lambda_\varphi$ can be used to define new measures of explainability generalizing the fraction of explained variance.

Ansari, J., P. B. Langthaler, S. Fuchs, and W. Trutschnig. Quantifying and estimating dependence via sensitivity of conditional distributions. Available at https://arxiv.org/abs/2308.06168.

It is known that the usual n out of n bootstrap fails for Chatterjee’s rank correlation. To remedy this, we present an m out of n bootstrap which is consistent for Chatterjee’s rank correlation whenever asymptotic normality of Chatterjee’s rank correlation can be established. Our bootstrap outperforms alternative estimation methods and approximates the limiting distribution in the Kolmogorov distance as well as the Wasserstein distance. We also present some results on the asymptotic normality of Chatterjee’s rank correlation, which is non-trivial to establish.

H. Dette and M. Kroll (2024): A Simple Bootstrap for Chatterjee’s Rank Correlation. Biometrika. DOI: 10.1093/biomet/asae045

M. Kroll (2024+): Asymptotic Normality of Chatterjee’s Rank Correlation. Preprint. arXiv: 2408.11547

Motivated by Chatterjee's rank correlation, we propose a novel rearrangement-invariant dependence order for conditional distributions that reflects many desirable properties of dependence measures. Our dependence order is based on the Schur-order for functions and capable to characterize independence of a random variable \(Y\) and a random vector \(\mathbf{X}\) as well as perfect dependence of \(Y\) on \(\mathbf{X}\,.\) Further, it satisfies an information gain inequality and is also able to characterizes conditional independence. As we show, our dependence order transfers all its fundamental properties to a large class of dependence measures. Moreover, it yields new supermodular comparison results for multi-factor models and thus various applications to robust risk models.