Ordinal sums and diagonal cuts

An ordinal sum places one copula in a lower diagonal block and another in an upper diagonal block. Let \(D,E\) be bivariate copulas and \(0<a<1\). Then

\[ (D\oplus_a E)(u,v)= \begin{cases} aD(u/a,v/a), & u,v\leq a,\\ a+(1-a)E\!\left(\frac{u-a}{1-a},\frac{v-a}{1-a}\right), & u,v\geq a,\\ \min(u,v), & \text{otherwise}. \end{cases} \]

The formulas agree on the block boundaries. Probabilistically, the lower block has mass \(a\) and the upper block has mass \(1-a\); within a chosen block, the corresponding component is affinely rescaled.

Recognizing an ordinal sum

Nelsen's binary converse ordinal-sum theorem

Let \(C\) be a bivariate copula and fix \(0<a<1\). Then

\[C(a,a)=a\iff\exists!\,(D,E)\text{ such that }C=D\oplus_a E.\]

Formal statementSource and proofProbabilityTheory.Copula.diagonal_eq_iff_existsUnique_ordinalSum

The component formulas are explicit:

\[ D(s,t)=\frac{C(as,at)}a,\qquad E(s,t)=\frac{C(a+(1-a)s,a+(1-a)t)-a}{1-a}. \]

The construction proves that these are copulas and that reassembling them recovers \(C\).

Formal statementSource and proofProbabilityTheory.Copula.ordinalSum_components

Uniqueness concerns the pair at the chosen split. It does not assert that the split itself is unique: \(M\) has a cut at every interior point.

The cut as a probability statement

Threshold disagreement

If \((U,V)\sim C\), then for every \(a\in I\),

\[ \mathbb P\bigl(\mathbf1_{\{U\leq a\}}\ne\mathbf1_{\{V\leq a\}}\bigr) =2\bigl(a-C(a,a)\bigr). \]

Formal statementSource and proofProbabilityTheory.Copula.measureReal_threshold_disagreement

Thus \(C(a,a)=a\) means that both coordinates fall on the same side of the threshold almost surely. No density is needed, and this identity includes \(a=0\) and \(a=1\).

If \(C\) is NQD, then \(C(a,a)\leq a^2<a\) for interior \(a\). Consequently an NQD copula cannot have a nontrivial binary ordinal-sum decomposition.

Formal statementSource and proofProbabilityTheory.Copula.IsNQD.not_exists_ordinalSum

Rank formulas

The formulas below hold for all \(a\in[0,1]\), with the degenerate sums \(D\oplus_0E=E\) and \(D\oplus_1E=D\):

\[ \tau(D\oplus_aE)=1-a^2(1-\tau(D))-(1-a)^2(1-\tau(E)), \]

\[ \rho(D\oplus_aE)=1-a^3(1-\rho(D))-(1-a)^3(1-\rho(E)), \]

\[ \phi(D\oplus_aE)=1-a^2(1-\phi(D))-(1-a)^2(1-\phi(E)). \]

Formal statementSource and proofProbabilityTheory.Copula.kendallTau_ordinalSum

Formal statementSource and proofProbabilityTheory.Copula.spearmanRho_ordinalSum

Formal statementSource and proofProbabilityTheory.Copula.spearmanFootrule_ordinalSum

The quadratic and cubic weights differ because the coefficients integrate different functions against different measures.

Maximal beta

An equal-split characterization

\[\beta(C)=1\iff\exists!\,(D,E)\text{ such that }C=D\oplus_{1/2}E.\]

Formal statementSource and proofProbabilityTheory.Copula.blomqvistBeta_eq_one_iff_existsUnique_ordinalSum

This characterization allows many copulas beyond \(M\). It also implies

\[\beta(C)=1\Longrightarrow\rho(C)\geq\tfrac12\quad\text{and}\quad\tau(C)\geq0.\]

Formal statementSource and proofProbabilityTheory.Copula.half_le_spearmanRho_of_blomqvistBeta_eq_one

Formal statementSource and proofProbabilityTheory.Copula.kendallTau_nonneg_of_blomqvistBeta_eq_one

The complete ordinal-sum guide includes sharpness examples, footrule bounds, beta \(-1\), component recovery, and tail behavior. Finite sums and countably many adjacent blocks with endpoints tending to one now have proved constructors; see the guide for their hypotheses. Arbitrary disjoint intervals with a residual comonotonic part and canonical decomposition into indecomposable components remain future work.

Library revision: fe53ea2f · Lean 4.34.0