arXiv subjects

Marcus Rockel

Publications and source records attributed to Marcus Rockel.

12 recordsLinked to original sources

arXiv2026

Exact regions for the symmetrized Blest coefficient

Spearman's rho treats all ranks alike, yet in many comparisons of two rankings an agreement at the top matters more than one at the bottom. Blest's rank correlation builds this preference into Spearman's rho, and its symmetrized version $η$, introduced by Genest and Plante, does so without favoring either of the two rankings. How far can such a top-weighted coefficient stray from other rank correlations? We answer this question for Spearman's rho, Blomqvist's beta and Chatterjee's rank correlation $ξ$ by determining the exact set of values that $η$ and each of these coefficients can attain jointly. Given Spearman's rho, $η$ can differ from it by at most $27/128$, and the extreme values of $η$ are attained only when the two rankings agree perfectly on the top ranks and are completely reversed on the bottom ranks, or vice versa. Given Blomqvist's beta, $η$ obeys exactly the same sharp bounds as Spearman's rho. For Chatterjee's rank correlation, both boundary curves are explicit and are attained by copulas with densities. Here symmetrization has a surprising effect. Compared with Blest's original coefficient, $η$ has a narrower range at fixed Spearman's rho, but near independence a wider range at fixed Chatterjee's rank correlation, where $|η|$ can come close to $\sqrt{11ξ/5}$ while Blest's coefficient stays below $\sqrt{32ξ/15}$. The proofs combine explicit Kantorovich potentials from optimal transport with a convex relaxation that turns out to be exact.

math.ST↗

arXiv2026

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

We determine the exact region $Ω_{β,ϕ,τ}= \{(β(C),ϕ(C),τ(C))\mid C\in\mathcal{C}\}$ of possible joint values of Blomqvist's beta, Spearman's footrule and Kendall's tau over the class $\mathcal{C}$ of all bivariate copulas. The region consists precisely of all triples $(β,ϕ,τ)$ satisfying $-1\le β\le 1$, $\frac{3}{16}(1+β)^2-\frac12\le ϕ\le 1-\frac38(1-β)^2$ and $\frac43 ϕ-\frac13\le τ\le \frac23 ϕ+\frac13$. In other words, the known exact $(β,ϕ)$- and $(ϕ,τ)$-regions already characterize the joint region, so that, once the value of Spearman's footrule is fixed, Blomqvist's beta imposes no additional restriction on the possible values of Kendall's tau. The proof is constructive: two $2$-parameter families of shuffles of $M$ realize all the points in $Ω_{β,ϕ,τ}$ with $ϕ= \frac{3}{16}(1+β)^2-\frac12$, and ordinal sums spread realizations of these two families over the whole region $Ω_{β,ϕ,τ}$.

math.ST↗

arXiv2026

The exact region between Chatterjee's $ξ$ and Blomqvist's $β$

We determine the exact region of Chatterjee's rank correlation $ξ$ and Blomqvist's $β$, that is, the set of all pairs $(ξ(C),β(C))$ where $C$ ranges over all bivariate copulas. The region equals $\{(x,y)\in[0,1]\times[-1,1]:|y|^3\le2x\}$, so every copula satisfies the sharp inequality $|β(C)|^3\le2ξ(C)$. Its left boundary $ξ=|β|^3/2$ is attained uniquely by an explicit family of copulas that perturb independence by a signed tent function centered at the median, and its right boundary $ξ=1$ by deterministic interval-exchange copulas. The proof reduces the problem to a one-dimensional minimization of an $L^2$-norm over $1$-Lipschitz functions. We further show that the region does not shrink when restricted to radially symmetric copulas and that positively quadrant dependent copulas attain exactly its upper half. For copulas under which the second variable is stochastically increasing in the first, the exact region is $\{(x,y)\in[0,1]^2:y^3\le2x,\ 3(1-y)^2\le4(1-x)\}$, and its right boundary is attained uniquely by explicit singular two-branch copulas. In particular, under stochastic increasingness $β=0$ is compatible with every value $ξ\in[0,1/4]$, whereas $ξ>1/4$ forces $β>0$.

math.ST↗

arXiv2026

Rank weighting and asymmetry in Blest's rank correlation: two exact regions

Blest's rank correlation $ν$ is a variant of Spearman's rho $ρ$ that weights the leading ranks of one variable more heavily, at the price that $ν$ is not symmetric in its arguments. We quantify both features by determining the exact region of $(ρ,ν)$ over all bivariate copulas, as well as that of $(η,ν)$, where $η$ is the symmetrized Blest coefficient of Genest and Plante. The latter region is a linear image of the set of all pairs $(ν(C),ν(C^\top))$ formed by a copula $C$ and its transpose. Consequently, Blest's coefficient differs from Spearman's rho by at most $1/4$, and interchanging the two variables changes it by at most $27/64$, improving on the bound $1/2$ implied by the first inequality. For every given value of $ρ$ or $η$, each corresponding extreme value of $ν$ is attained by exactly one copula, given in closed form and supported on finitely many line segments. Near countermonotonicity, the upper extremizers of the $(η,ν)$-region are supported on the graph of a function of the second coordinate, yet their conditional laws given the first coordinate carry two atoms. The proofs rest on a rearrangement inequality with equality case and on explicit Kantorovich potentials.

math.ST↗

arXiv2026

The exact region determined by Spearman's rho and Gini's gamma

We determine the exact attainable region of Spearman's rho and Gini's gamma over all bivariate copulas, resolving the remaining pairwise exact-region problem among Spearman's rho, Kendall's tau, Gini's gamma, Blomqvist's beta and Spearman's footrule. We give an explicit parametrization of the rho-maximal boundary for prescribed Gini's gamma and construct copulas attaining every boundary point. The boundary consists of an elementary arc up to a single junction and, beyond it, countably many algebraic pieces accumulating at comonotonicity. This description also gives the largest possible value of $|ρ-γ|$ and the sharp thresholds beyond which rho and gamma must have the same sign. Our proof separates the signs and magnitudes of centred ranks, reducing the optimization of linear combinations of $ρ$ and $γ$ to an optimal transport problem for two uniform magnitudes. The extremizers combine a central antidiagonal block with rescaled auxiliary transport optimizers, and global optimality follows from a Kantorovich dual potential obtained by gluing the corresponding dual pieces.

math.ST↗

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 $ξ(X,Y)$, 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.

math.ST↗

arXiv2026

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

We compare Chatterjee's rank correlation $ξ$ with Kendall's $τ$ and Spearman's $ρ$ 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 $ξ(C)\leq τ(C)$ 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 $ξ(C)\leq ρ(C)$, 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 $ξ(C)\leqρ(C)$, that LTD and RTI together do not imply $ξ(C)\leqτ(C)$, and that both bounds are directional for $ξ$.

math.ST↗

arXiv2026

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 $ξ$, 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 $ξ(C_n) \le ξ(C)$ with $ξ(C_n) \to ξ(C)$ as $n\to\infty$.

math.ST↗

arXiv2026

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

The rank correlation ξ(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 ρ, which capture the degree of positive or negative dependence, ξquantifies the strength of functional dependence. In this paper, we study the attainable set of pairs (ξ(X,Y),ρ(X,Y)). The resulting ξ-\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 ξ(X,Y)\leq|ρ}(X,Y)| whenever Y is stochastically increasing or decreasing in X, and we identify the maximal difference ρ(X,Y)-ξ(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 ξand ρ. Our results contribute to a better understanding of Chatterjee's rank correlation, which typically yields substantially smaller values than Spearman's ρwhen quantifying positive dependencies. In particular, when interpreting the values of Chatterjee's rank correlation on the scale of ρ, the quantity \sqrtξ appears to be more appropriate.

math.ST↗

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 $ξ$ and Blest's rank correlation $ν$ 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 $ξ$ and $ν$, which provide an explicit parametrization of the exact attainable region.

math.ST↗

arXiv2025

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

Chatterjee's rank correlation $ξ$ 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, $ξ$ equals Spearman's footrule~$ψ$ 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 $(ξ, ψ)$ over all bivariate copulas. In particular, we show that for given $ξ$, the maximal possible value of $ψ$ is uniquely attained by a Fréchet copula. As a by-product of this and a known result for Markov products of copulas, we obtain that $ξ\leψ\le \sqrtξ$ characterizes the exact region of stochastically increasing copulas. Regarding the minimal possible value of $ψ$ for given $ξ$, we give a lower bound based on Jensen's inequality and construct a two-parameter copula family that comes comparably close.

math.ST↗

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.

math.ST↗