arXiv subjects

Marcus Rockel

Publications and source records attributed to Marcus Rockel.

9 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

The exact region determined by Kendall's tau, Spearman's footrule and Blomqvist's beta

We determine the exact region $\Omega_{\tau,\phi,\beta}:=\{(\tau(C),\phi(C),\beta(C)):C\in\mathcal{C}\}$ of possible joint values of Kendall's tau, Spearman's footrule and Blomqvist's beta over the class $\mathcal{C}$ of all bivariate copulas. The region consists precisely of all triples $(t,p,b)$ satisfying $-1\le b\le 1$, $\frac{3}{16}(1+b)^2-\frac12\le p\le 1-\frac38(1-b)^2$ and $\frac43 p-\frac13\le t\le \frac23 p+\frac13$. In other words, the known exact $(\phi,\beta)$- and $(\tau,\phi)$-regions already characterize the joint region, so that, once the value of Spearman's footrule is fixed, Blomqvist's beta imposes no additional sharp restriction on the possible values of Kendall's tau. The proof is constructive: two one-parameter families of shuffles of $M$ realize the extreme values of Kendall's tau along the lower boundary of the $(\phi,\beta)$-region, ordinal sums spread these families through the whole region, and the vertical fibres are filled using the biaffinity of the concordance function. We further show that $\Omega_{\tau,\phi,\beta}$ is convex with rectangular fixed-footrule sections, identify an affine symmetry of its fibres about $\tau=\phi$, and compute its volume, which equals $\frac{31}{40}$.

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arXiv2026

The exact region between Chatterjee's $\xi$ and Blomqvist's $\beta$

We determine the exact attainable region of the pair $(\xi(C),\beta(C))$ formed by Chatterjee's rank correlation $\xi$ and Blomqvist's $\beta$ over the class of all bivariate copulas and show that it is given by $\{(x,y)\in[0,1]\times[-1,1]: |y|^3\le 2x\}.$ The left boundary $\xi=|\beta|^3/2$ is attained by an explicit two-strip family $(L_b)_{b\in[-1,1]}$ obtained by perturbing independence with a signed tent function $g_b$ centered at the median. We derive several properties of this copula family including the formulas for its density and rank correlation measures, as well as positive and negative dependence properties. The right boundary $\xi=1$ is attained for every admissible value of $\beta$ by deterministic measure-preserving copulas, and the full region is obtained by taking convex mixtures of the left- and right-boundary copulas with fixed $\beta$ and using the continuity of $\xi$ along these mixtures. We also record the exact regions in several natural subclasses of copulas.

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arXiv2026

Kendall and Spearman bounds for Chatterjee's rank correlation under positive dependence

We compare Chatterjee's rank correlation $\xi$ with Kendall's $\tau$ and Spearman's $\rho$ under positive-dependence assumptions on bivariate copulas. Our main technical contribution is a sharp order-violation bound for two stochastically ordered distribution functions. This local inequality controls each conditional order-violation probability appearing in Kendall's tau by the cross-rank variance functionals that determine Chatterjee's rank correlation. As a consequence, we prove the sharp Kendall bound $\xi(C)\leq \tau(C)$ for every stochastically increasing copula $C$. The bound is best possible: ordinal sums of product copulas attain equality. We also prove that the weaker left-tail decreasing (LTD) and right-tail increasing (RTI) conditions jointly imply the Spearman bound $\xi(C)\leq \rho(C)$, with equality if and only if $C$ is either the independence or comonotonicity copula. Finally, checkerboard examples show that LTD or RTI alone does not imply $\xi(C)\leq\rho(C)$, that LTD and RTI together do not imply $\xi(C)\leq\tau(C)$, and that both bounds are directional for $\xi$.

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arXiv2026

The exact region between Chatterjee's and Blest's rank correlations

Exact regions between rank correlations describe the set of all pairs of values that two dependence measures can attain simultaneously on the same copula and thus yield sharp inequalities between them. In this paper, we determine the exact region between Chatterjee's rank correlation $\xi$ and Blest's rank correlation $\nu$ over the class of all bivariate copulas. Our approach is based on a constrained optimization problem whose solution is characterized by Karush--Kuhn--Tucker conditions. This leads to a novel extremal copula family that uniquely traces the boundary of the region. For this family, we derive closed-form expressions for both $\xi$ and $\nu$, which provide an explicit parametrization of the exact attainable region.

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arXiv2025

On the exact region between Chatterjee's rank correlation and Spearman's footrule

Chatterjee's rank correlation $\xi$ has emerged as a popular measure quantifying the strength of directed functional dependence between random variables $X$ and $Y$. If $X$ and $Y$ are continuous, $\xi$ equals Spearman's footrule~$\psi$ for the Markov product of the copula induced by $(X,Y)$ and its transpose. We analyze the relationship between these two measures more in depth by studying the attainable region of possible pairs $(\xi, \psi)$ over all bivariate copulas. In particular, we show that for given $\xi$, the maximal possible value of $\psi$ is uniquely attained by a Fr\'echet copula. As a by-product of this and a known result for Markov products of copulas, we obtain that $\xi\le\psi\le \sqrt{\xi}$ characterizes the exact region of stochastically increasing copulas. Regarding the minimal possible value of $\psi$ for given $\xi$, we give a lower bound based on Jensen's inequality and construct a two-parameter copula family that comes comparably close.

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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

Measures of association for approximating copulas

This paper studies closed-form expressions for multiple association measures of copulas commonly used for approximation purposes, including Bernstein, shuffle--of--min, checkerboard and check--min copulas. In particular, closed-form expressions are provided for the recently popularized Chatterjee's $\xi$, which quantifies the dependence between two random variables. Given an absolutely continuous bivariate copula $C$ with TP$_2$ density and approximating $n\times n$-checkerboard copula $C_n$, we show that $\xi(C_n) \le \xi(C)$ with $\xi(C_n) \to \xi(C)$ as $n\to\infty$.

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arXiv2024

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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