Attainable regions

Bounds, extremizers, and what an exact region requires.

What does an exact region mean?

Given two real-valued functionals $a$ and $b$ on a class $\mathcal C$ of copulas, their attainable region is

An inequality gives an outer constraint. An exact description additionally requires attainment: every point in the proposed set must come from an admissible copula. A boundary formula alone does not settle the interior.

The proof obligations

An article’s region theorem typically separates into several tasks:

1. Universal bounds. Show that each admissible copula produces a point in the proposed set.
2. Extremal construction. Define copulas attaining boundary values, and prove that they have uniform margins.
3. Attainment throughout. Construct a copula for every remaining point in the set.
4. Equality cases. State and prove any uniqueness claim with all exceptional and endpoint cases.

For three coefficients, the same logic applies in $\mathbb R^3$. Sections at a fixed coefficient can organize the argument, but their compatibility still needs proof.

A guide to the xi–rho article

The xi–rho supplement tracks three targets from arXiv v3: the attainable region (Theorem 1), the sharp maximum of $\rho-\xi$ (Corollary 1), and the inequality under stochastic monotonicity (Theorem 2). All three targets are now verified, including every equality case. The full curved boundary uses the source's explicit trigonometric and radical inverse formulas; every interior point is attained by an actual copula, and every interior boundary copula is unique. The source family's symmetry, parameter ordering, uniform limits, density, convex support, MTP2, asymmetry, and all three rank formulas are checked. This completes its corrected stated scope. The journal equation (19) omits a boundary term restored in arXiv v3; Lean proves both the corrected formula and a counterexample to the literal journal expression.

The paper proves $\xi(C)\leq|\rho(C)|$ for the stated stochastically increasing or decreasing classes. Keeping this class restriction attached to the inequality is essential. The verification proves that equality holds exactly at countermonotonicity, independence, or comonotonicity, without a density assumption. The supporting scalar Lemma 8 is checked with equality of functions interpreted almost everywhere. See the versioned article for the exact statements and equality cases.

Equality in the SI xi–footrule region

The xi–footrule supplement verifies the entire SI region and Proposition 2.2's full equality criterion for its lower boundary. Equality holds precisely when the conditional CDF has the source's three-level representation with measurable, nondecreasing cut functions. Both directions and the derivative convention are checked, including singular copulas. Theorem 3.2's universal Jensen lower bound is checked with the piecewise scalar optimizer and its uniqueness proof. Proposition 3.1 now provides exact rational and logarithmic coefficient formulas, continuity, strict monotonicity, and both endpoint values.

For target footrule in [-1/2,0], the inverse parameter exists uniquely in [0,2] and satisfies the source's cubic. This gives the explicit xi lower estimate in Theorem 3.3 on that range. The interval restriction is essential: at footrule -1/2 the cubic also has the inadmissible real root -1, besides 2. The coverage map records this correction to the source's global uniqueness wording. The relaxed profile also fails copula monotonicity, so no copula attainment is asserted for this estimate. Remaining region geometry and copula constructions retain their pending entries.

Convexity in three xi regions

The xi-footrule, xi-rho, and xi-Blest supplements now verify convexity of their full attainable regions. The proof constructs a copula for every convex combination of two attained pairs: first mix the given copulas, then mix with a xi=1 witness at the same second coefficient to reach the desired xi. The continuous xi path and affine second coefficient justify both steps. Closedness and the remaining curved-boundary formulas are separate obligations.

For xi-footrule, the entire xi=1 boundary is now checked as well. Centered countermonotonic blocks give every footrule value in [-1/2,1]. Together with the Frechet lower-xi endpoint at nonnegative footrule, these witnesses prove the exact portion 0<=footrule<=1 and footrule^2<=xi<=1, including attainment throughout. At negative footrule, the proved Jensen bound remains an outer estimate; convexity does not make it an attaining curve.

A complete three-coefficient region

The tau–footrule–beta supplement now verifies both directions of the full joint-region theorem. Explicit signed shuffles supply both tau endpoints at every admissible footrule/beta pair; continuous mixtures then attain every intermediate tau. The general signed shuffle formula includes zero-width strips and singular copulas.

The actual attained region is compact and convex, with the exact rectangular fixed-footrule sections and the affine fibre symmetry stated in the article. Its Lebesgue volume is 31/40. The fixed-beta section areas, their derivative, and their unique attained maximum are checked as well. The coverage map has no remaining pending results in its stated scope.

A picture is a guide, not a coverage claim

Plots and numerical optimizers can suggest extremizers. A machine-checked region statement still needs exact definitions and quantified proofs of the inclusions above. A finite sample of attainable points cannot establish attainment of a continuum.

Mixtures require care too. Even if copulas form a convex class, a coefficient may be nonlinear in the copula. A straight segment in coefficient space does not follow merely by mixing two copulas.

Find the corresponding supplement

Read each coverage map for its actual scope. In particular, an article title containing “exact region” is not a claim that its full result is verified here.

Full rho–footrule and rho–gamma regions

The latest exact-region additions in the copula package are now integrated into both article supplements, including singular copulas and all endpoints.

The rho–footrule supplement proves both sharp boundaries and every intervening point, converts the upper arcs to the source radical formulas, and proves the sharp attained variance correction. Its zero set is exactly zero and the means 1/(2N) for positive integers N. The exact mean–variance region and its universal maximum, finite-ranking inequalities, attained generalized mixability infimum, and xi/correlation-ratio outer bounds are also checked. The finite-ranking equality criterion and asymptotic sharpness, conditional-copy examples, full inner enclosure, and compactness and attained upper maxima of the entire xi-eta region are verified. Every upper-boundary point has a unique optimizing copula, including all junctions and the endpoint. The supplement is complete for its stated main-result scope.

The rho–gamma supplement proves the full exact region, compactness, and continuity, concavity and strict increase of the upper boundary. The elementary arc, exact largest discrepancy, and both sharp sign thresholds are checked. The supplement is now complete for its stated scope: the original theta family, all junctions and endpoint limits, full parameter coverage, exact five-piece graph laws, the uniform cubic endpoint expansion and attained strong duality for every positive multiplier are verified.