arXiv subjects

Jonathan Ansari

Publications and source records attributed to Jonathan Ansari.

13 recordsLinked to original sources

arXiv2026

The exact Spearman rho-footrule region via optimal transport with applications to finite rankings, mixability, and Chatterjee's rank correlation

We solve the open problem of determining the maximal value of Spearman's rho when Spearman's footrule is prescribed, thereby completing the exact attainable region of these two quantities. To prove this result, we reformulate the underlying copula optimization problem as an optimal transport problem with a linear moment constraint and construct the unique optimal coupling through a matching feasible dual potential and its contact set. Equivalently, this coupling minimizes the variance of $|U-V|$ among all couplings of $U,V\sim\mathcal{U}(0,1)$ with prescribed mean $\mathbb{E}|U-V|$. Our main result admits several applications: First, in the context of finite rankings, we obtain an improved Cauchy--Schwarz inequality between Spearman's footrule distance and the associated quadratic rank difference. Second, in the framework of generalized mixability, we characterize the attainable constant values of $|U'+V'|$, for $U',V'\sim\mathcal{U}(-1/2,1/2)$, and determine the minimal quadratic deviation for a given mean. Third, we derive explicit bounds relating Chatterjee's rank correlation $\xi(X,Y)$, which can detect complex functional dependence of $Y$ on $X$, to the copula correlation ratio--a rank-based fraction of explained variance--by exploiting their conditional i.i.d. representations in terms of Spearman's footrule and Spearman's rho.

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arXiv2026

Dependence Measures via Adapted Optimal Transport: Stability and Rates of Convergence

Recently studied dependence measures, such as Chatterjee's rank correlation, that characterize both independence and perfect functional dependence, provide a powerful framework for detecting nonlinear dependencies. However, these measures cannot be weakly continuous, which limits the applicability of classical plug-in estimators based on empirical distributions. This obstruction is natural, as such measures are defined via conditional distributions and not through their joint law alone. In this paper, we introduce an optimal transport-based mode of convergence that captures weak convergence of conditional distributions and restores continuity for a broad class of dependence measures. We relate this mode of convergence to the adapted Wasserstein distance, the Knothe-Rosenblatt distance and the d1-metric on copulas. Building on this perspective, we propose a copula estimator based on the adapted empirical measure and compare it with the classical rank-based checkerboard estimator. For both estimators, we derive O(N^{-1/3})-rates of convergence with respect to metrics that capture conditional weak continuity. As a consequence, we obtain the same rates for plug-in estimators of several classes of dependence measures, including rank-based and rearranged dependence measures.

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arXiv2026

On a copula product characterizing independence and perfect functional dependence

Recently studied dependence measures take values in [0,1], where 0 characterizes independence of X and Y, and 1 characterizes perfect functional (not necessarily monotone) dependence of Y on X. The most prominent example is Chatterjee's rank correlation, which is based on the concept of conditional independence. In contrast, we show that dependence measures such as Wasserstein correlations and rearranged dependence measures are naturally linked to the concept of conditional comonotonicity. This connection is captured by a new copula product T(C)=C\vee\Pi which models conditional comonotonicity and characterizes independence and perfect functional dependence. As a main contribution, we prove that the mapping T acts as a reflection on the class of stochastically increasing copulas. Further, T^2=T \circ T projects a copula onto its increasing rearranged copula. To better understand the behavior of such dependence measures, we also study fixed points, ordering results, and continuity properties of T. Our results demonstrate that conditional comonotonicity is a rather intrinsic feature of dependence measures, whereas conditional independence underlying Chatterjee's rank correlation is a quite exceptional property.

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arXiv2025

An ordering for the strength of functional dependence

We introduce a new dependence order, termed the conditional convex order, whose minimal and maximal elements characterize independence and perfect dependence. Moreover, it characterizes conditional independence, satisfies information monotonicity, and exhibits several invariance properties. Consequently, it is an ordering for the strength of functional dependence of a random variable Y on a random vector X. As we show, various recently studied dependence measures -- including Chatterjee's rank correlation, Wasserstein correlations, and rearranged dependence measures -- are increasing in this order and inherit their fundamental properties from it. We characterize the conditional convex order by the Schur order and by the concordance order, and we verify it in settings such as additive error models, the multivariate normal distribution, and various copula-based models. Our results offer a unified perspective on the behavior of dependence measures across statistical models.

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arXiv2025

The exact region and an inequality between Chatterjee's and Spearman's rank correlations

The rank correlation \xi(X,Y), recently established by Sourav Chatterjee and already popular in the statistics literature, takes values in [0,1], where 0 characterizes independence of X and Y, and 1 characterizes perfect dependence of Y on X. Unlike concordance measures such as Spearman's \rho, which capture the degree of positive or negative dependence, \xi quantifies the strength of functional dependence. In this paper, we study the attainable set of pairs (\xi(X,Y),\rho(X,Y)). The resulting {\xi}-\r{ho}-region is a convex set whose boundary is characterized by a novel family of absolutely continuous, asymmetric copulas having a diagonal band structure. Moreover, we prove that \xi(X,Y)\leq|\rho}(X,Y)| whenever Y is stochastically increasing or decreasing in X, and we identify the maximal difference \rho(X,Y)-\xi(X,Y) as exactly 0.4. Our proofs rely on a convex optimization problem under various equality and inequality constraints, as well as on ordering properties for \xi and \rho. Our results contribute to a better understanding of Chatterjee's rank correlation, which typically yields substantially smaller values than Spearman's \rho when quantifying positive dependencies. In particular, when interpreting the values of Chatterjee's rank correlation on the scale of \rho, the quantity \sqrt{\xi} appears to be more appropriate.

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arXiv2025

On continuity of Chatterjee's rank correlation and related dependence measures

While measures of concordance -- such as Spearman's rho, Kendall's tau, and Blomqvist's beta -- are continuous with respect to weak convergence, Chatterjee's rank correlation xi recently introduced in Azadkia and Chatterjee (2021) does not share this property, causing drawbacks in statistical inference as pointed out in B\"ucher and Dette (2025). As we study in this paper, xi is instead weakly continuous with respect to conditionally independent copies -- the Markov products. To establish weak continuity of Markov products, we provide several sufficient conditions, including copula-based criteria and conditions relying on the concept of conditional weak convergence in Sweeting (1989). As a consequence, we also obtain continuity results for xi and related dependence measures and verify their continuity in the parameters of standard models such as multivariate elliptical and l1-norm symmetric distributions.

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arXiv2024

Robust Bernoulli Mixture Models for Credit Portfolio Risk

This paper presents comparison results and establishes risk bounds for credit portfolios within classes of Bernoulli mixture models, assuming conditionally independent defaults that are stochastically increasing with a common risk factor. We provide simple and interpretable conditions for conditional default probabilities that imply a comparison of credit portfolio losses in convex order. In the case of threshold models, the ranking of portfolio losses is based on a pointwise comparison of the underlying copulas. Our setting includes as special case the well-known Gaussian copula model but allows for general tail dependencies, which are crucial for modeling credit portfolio risks. Moreover, our results extend the classical parameterized models, such as the industry models CreditMetrics and KMV Portfolio Manager, to a robust setting where individual parameters or the copula modeling the dependence structure can be ambiguous. A simulation study and a real data example under model uncertainty offer evidence supporting the effectiveness of our approach.

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arXiv2024

Comparison results for positive supermodular dependent Markov tree distributions

Positive dependencies have been compared in the literature under rather strong assumptions such as equality of conditional distributions, exchangeability, or stationarity. We establish supermodular ordering results for distributions that are Markov with respect to a tree structure. Our comparison results rely on simple stochastic monotonicity conditions and a pointwise ordering of bivariate copulas associated with the edges of the underlying tree. We also study flexibility of the marginal distributions in stochastic and convex order. As a consequence, we obtain first- and second-order stochastic dominance esults for extreme order statistics and sums of positively dependent random variables. As an application, we investigate distributional robustness of the maximum of a perturbed random walk under model uncertainty. Several examples and a detailed discussion of the assumptions demonstrate the generality of our results and reveal deeper insights into non-intuitive positive dependence properties of multidimensional distributions.

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arXiv2023

Dependence properties of bivariate copula families

Motivated by recently investigated results on dependence measures and robust risk models, this paper provides an overview of dependence properties of many well-known bivariate copula families, where the focus is on the Schur order for conditional distributions, which has the fundamental property that minimal elements characterize independence and maximal elements characterize perfect directed dependence. We give conditions on copulas that imply the Schur ordering of the associated conditional distribution functions. For extreme-value copulas, we prove the equivalence of the lower orthant order, the Schur order for conditional distributions, and the pointwise order of the associated Pickands dependence functions. Further, we provide several tables and figures that list and illustrate various positive dependence and monotonicity properties of copula families, in particular, from classes of Archimedean, extreme-value, and elliptical copulas. Finally, for Chatterjee's rank correlation, which is consistent with the Schur order for conditional distributions, we give some new closed-form formulas in terms of the parameter of the underlying copula family.

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arXiv2023

Quantifying and estimating dependence via sensitivity of conditional distributions

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 marginal distribution of $Y$. They assign pairs $(X,Y)$ values in $[0,1]$, the value is $0$ if and only if $X,Y$ are independent, and it is $1$ exclusively for $Y$ being a 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 construction underlying $\Lambda_\varphi$ can be used to define new measures of explainability generalizing the fraction of explained variance.

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arXiv2022

A direct extension of Azadkia & Chatterjee's rank correlation to multi-response vectors

Recently, Chatterjee (2023) recognized the lack of a direct generalization of his rank correlation $\xi$ in Azadkia and Chatterjee (2021) to a multi-dimensional response vector. As a natural solution to this problem, we here propose an extension of $\xi$ that is applicable to a set of $q \geq 1$ response variables, where our approach builds upon converting the original vector-valued problem into a univariate problem and then applying the rank correlation $\xi$ to it. Our novel measure $T$ quantifies the scale-invariant extent of functional dependence of a response vector $\mathbf{Y} = (Y_1,\dots,Y_q)$ on predictor variables $\mathbf{X} = (X_1, \dots,X_p)$, characterizes independence of $\mathbf{X}$ and $\mathbf{Y}$ as well as perfect dependence of $\mathbf{Y}$ on $\mathbf{X}$ and hence fulfills all the characteristics of a measure of predictability. Aiming at maximum interpretability, we provide various invariance results for $T$ as well as a closed-form expression in multivariate normal models. Building upon the graph-based estimator for $\xi$ in Azadkia and Chatterjee (2021), we obtain a non-parametric, strongly consistent estimator for $T$ and show -- as a main contribution -- its asymptotic normality. Based on this estimator, we develop a model-free and rank-based feature ranking and forward feature selection for multiple-outcome data that works without any tuning parameters. Simulation results and real case studies illustrate $T$'s broad applicability.

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arXiv2022

Improved Robust Price Bounds for Multi-Asset Derivatives under Market-Implied Dependence Information

We show how inter-asset dependence information derived from market prices of options can lead to improved model-free price bounds for multi-asset derivatives. Depending on the type of the traded option, we either extract correlation information or we derive restrictions on the set of admissible copulas that capture the inter-asset dependencies. To compute the resultant price bounds for some multi-asset options of interest, we apply a modified martingale optimal transport approach. Several examples based on simulated and real market data illustrate the improvement of the obtained price bounds and thus provide evidence for the relevance and tractability of our approach.

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arXiv2022

On a version of a multivariate integration by parts formula for Lebesgue integrals

Multidimensional integration by parts formulas apply under the standard assumption that one of the functions is continuous and the other has bounded Hardy-Krause variation. Motivated by recently developed results in the probabilistic context of price and risk bounds, this paper provides a version of an integration by parts formula for the Lebesgue integral of measure-inducing functions which may both be discontinuous and may have infinite Hardy-Krause variation. To this end, we give a general definition of measure-inducing functions and establish various of their properties, such as a characterization in terms of Delta-monotone functions. As a consequence of the integration by parts formula, several convergence results are provided, allowing an extension of the Lebesgue integral of a measure-inducing function to the case where one integrates with respect to a continuous semi-copula. The latter class of aggregation functions includes quasi-copulas which serve as bounds for the dependence structure in many applications.

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