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arxiv: 2606.30033 · v1 · pith:W4CKTQX4new · submitted 2026-06-29 · 🧮 math.ST · stat.TH

The exact region between Chatterjee's xi and Blomqvist's β

Jacob Israel Orenday Lares , Marcus Rockel This is my paper

Pith reviewed 2026-06-30 04:05 UTC · model grok-4.3

classification 🧮 math.ST stat.TH
keywords Chatterjee rank correlationBlomqvist betabivariate copulasattainable regionrank correlation measuresdependence propertiescopula mixtures
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The pith

Chatterjee's ξ and Blomqvist's β for any bivariate copula satisfy exactly |β|^3 ≤ 2ξ.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the precise set of attainable pairs (ξ(C), β(C)) over all bivariate copulas C. It proves this set equals the region defined by the inequality |y|^3 ≤ 2x. The lower boundary is realized by an explicit family of copulas that add a signed tent perturbation to independence, while the upper boundary ξ=1 holds for deterministic copulas at every admissible β. Convex mixtures of these boundary families, combined with continuity of ξ, fill the entire region.

Core claim

We determine the exact attainable region of the pair (ξ(C),β(C)) formed by Chatterjee's rank correlation ξ and Blomqvist's β over the class of all bivariate copulas and show that it is given by {(x,y)∈[0,1]×[-1,1]: |y|^3≤2x}. The left boundary ξ=|β|^3/2 is attained by an explicit two-strip family (L_b) obtained by perturbing independence with a signed tent function g_b centered at the median. The right boundary ξ=1 is attained for every admissible value of β by deterministic measure-preserving copulas, and the full region is obtained by taking convex mixtures of the left- and right-boundary copulas with fixed β and using the continuity of ξ along these mixtures.

What carries the argument

The two-strip family (L_b) of copulas, formed by perturbing the independence copula with a signed tent function g_b centered at the median, which attains the left boundary ξ = |β|^3/2.

If this is right

  • The left boundary ξ = |β|^3/2 is attained exactly by the explicit two-strip family L_b.
  • Deterministic measure-preserving copulas attain ξ = 1 for every admissible value of β.
  • Convex mixtures of left- and right-boundary copulas with the same β fill the interior of the region.
  • The density, rank correlations, and positive/negative dependence properties of the L_b family are given by explicit formulas.
  • Exact attainable regions are also recorded for several natural subclasses of copulas.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The cubic boundary may guide construction of new dependence measures whose joint range is easier to characterize.
  • Similar mixture arguments could determine attainable regions for other pairs of rank correlations such as Spearman's rho paired with β.
  • In applications, observed pairs (ξ̂, β̂) lying outside the region would indicate either estimation error or model misspecification.
  • The continuity of ξ under convex mixing suggests that the map from copulas to (ξ, β) is continuous in the weak topology for fixed β.

Load-bearing premise

The continuity of ξ along convex mixtures of the left-boundary family and right-boundary deterministic copulas with fixed β.

What would settle it

A single bivariate copula C for which |β(C)|^3 > 2 ξ(C), or a point inside |y|^3 < 2x shown to be unattainable by any copula.

Figures

Figures reproduced from arXiv: 2606.30033 by Jacob Israel Orenday Lares, Marcus Rockel.

Figure 1
Figure 1. Figure 1: The exact region Rξ,β of Chatterjee’s ξ and Blomqvist’s β over all bivariate copulas (Theorem 1). The curved left boundary is the cubic ξ = |β| 3/2, at￾tained uniquely by the tent copulas Lb. Further, the straight right boundary ξ = 1 is attained by the deter￾ministic measure-preserving copulas Db. Marked are the independence copula Π(u, v) = uv, the Fr´echet– Hoeffding bounds M(u, v) = min{u, v} and W(u, … view at source ↗
Figure 2
Figure 2. Figure 2: The density cb of the boundary copula Lb, see (7), for b = 0.4, 0.7, and 1. It is piecewise constant on rectangular blocks and takes only the values 0, 1 (light), and 2 (dark). Solid lines mark the marginal medians u = 1 2 and v = 1 2 , dashed lines the edges v = αb, γb of the tent support. As b ↑ 1, the density-2 blocks grow until, at b = 1, the copula is supported, with density 2, on the two diagonal med… view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper determines the exact attainable region of the pair (Chatterjee's ξ(C), Blomqvist's β(C)) over all bivariate copulas C, showing that it equals the set {(x,y) ∈ [0,1] × [-1,1] : |y|^3 ≤ 2x}. The left boundary ξ = |β|^3/2 is attained by an explicit two-strip family (L_b) obtained by perturbing independence with a signed tent function g_b centered at the median; the right boundary ξ=1 is attained for each admissible β by deterministic measure-preserving copulas; the interior is filled by convex mixtures of these families (with fixed β) together with a continuity argument for ξ along the mixture paths.

Significance. If the central claim holds, the result supplies a sharp, explicit description of the joint range of two well-known rank correlations, clarifying their relationship in a way that is useful for theoretical work on dependence measures and for interpreting empirical values. The explicit construction of the boundary families, the derivation of their densities and rank-correlation formulas, and the recording of attainable regions within natural subclasses (e.g., absolutely continuous copulas) constitute concrete, reusable contributions.

major comments (1)
  1. [abstract (final sentence of main-result paragraph) / section describing the mixtures] The exact-region claim rests on the assertion that, for each fixed b, the convex path C_t = (1-t) L_b + t D_b (t ∈ [0,1]) produces ξ(C_t) values that continuously fill the entire interval [|b|^3/2, 1]. The abstract invokes continuity of ξ along these mixtures to conclude that the region is filled, but the justification (direct verification of the integral expression for ξ under the mixture measure, or an appeal to dominated convergence or uniform continuity) is load-bearing; if this step is only sketched, the attainable set could be strictly smaller than claimed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for a fully rigorous justification of the continuity step. We address the single major comment below and will incorporate the requested detail in the revision.

read point-by-point responses

  1. Referee: [abstract (final sentence of main-result paragraph) / section describing the mixtures] The exact-region claim rests on the assertion that, for each fixed b, the convex path C_t = (1-t) L_b + t D_b (t ∈ [0,1]) produces ξ(C_t) values that continuously fill the entire interval [|b|^3/2, 1]. The abstract invokes continuity of ξ along these mixtures to conclude that the region is filled, but the justification (direct verification of the integral expression for ξ under the mixture measure, or an appeal to dominated convergence or uniform continuity) is load-bearing; if this step is only sketched, the attainable set could be strictly smaller than claimed.

    Authors: We agree that an explicit verification of continuity is load-bearing and should be expanded. In the revised manuscript we will add a short subsection that writes the integral formula for ξ(C) (the one appearing in Chatterjee (2021) or the equivalent expression in terms of the copula measure) and applies the dominated-convergence theorem directly to the mixture densities. Because the densities of L_b and D_b are bounded and the mixture densities converge pointwise and are dominated by an integrable function independent of t, ξ(C_t) is continuous in t. Combined with the already-established endpoint values ξ(L_b)=|b|^3/2 and ξ(D_b)=1, the intermediate-value theorem then guarantees that the image is the full interval. This change removes any ambiguity about whether the attainable set might be strictly smaller. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit constructions and continuity argument

full rationale

The derivation constructs an explicit left-boundary family L_b via signed tent perturbation of independence, attains the right boundary ξ=1 via deterministic copulas, and fills the region via convex mixtures C_t = (1-t)L_b + t D_b while invoking continuity of ξ along these paths. All steps rely on direct copula definitions, density formulas, and measure-theoretic continuity rather than any self-definition, fitted-parameter renaming, or load-bearing self-citation. The argument is self-contained and does not reduce the target region to quantities defined in terms of itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper works entirely within the standard theory of bivariate copulas and introduces one explicit parametric family to attain the boundary; no free parameters are fitted to data and no new entities with independent evidence are postulated.

axioms (1)
  • standard math Standard properties of bivariate copulas (grounded, 2-increasing, uniform margins)
    The entire development is carried out inside the class of all bivariate copulas as defined in the literature.
invented entities (1)
  • two-strip family (L_b) with signed tent perturbation g_b no independent evidence
    purpose: To attain the left boundary ξ = |β|^3 / 2
    Explicitly constructed in the paper by perturbing the independence copula; no independent evidence outside the construction is given.

pith-pipeline@v0.9.1-grok · 5738 in / 1331 out tokens · 60059 ms · 2026-06-30T04:05:19.759037+00:00 · methodology

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Reference graph

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