Orders and positive dependence¶
Different comparisons answer different questions. This chapter uses bivariate copulas and the orientation second coordinate given first for directional dependence properties.
Lower orthant and concordance order¶
Lower orthant order compares the distribution functions pointwise:
Concordance order compares both lower-orthant and upper-orthant probabilities. In two dimensions, these conditions coincide with the pointwise comparison above.
Strict monotonicity of rho among comparable copulas
If \(C\preceq_{\mathrm{lo}}D\) and \(C\ne D\), then
Comparability is essential. Rho is a single number and does not identify an arbitrary copula among all copulas.
Schur order through conditional CDFs¶
For \(K_C(u,t)=\mathbb P(V\leq t\mid U=u)\), the library uses the convex-test formulation
for every \(t\in I\) and every continuous \(\psi:\mathbb R\to\mathbb R\) that is convex on \(I\). Each conditional section has mean \(t\).
Independence is least
Every bivariate copula satisfies \(\Pi\preceq_{\mathrm{Schur}}C\).
Xi respects Schur order
Schur order is a preorder here: distinct copulas can be comparable in both directions. For the exact conventions and the scope of the formalization, see the orders guide.
Positive dependence implications¶
For each fixed \(v\), the principal directional conditions are:
| Condition | Mathematical meaning |
|---|---|
| PQD | \(C(u,v)\geq uv\) |
| LTD | \(u\mapsto C(u,v)/u\) is nonincreasing on \((0,1]\) |
| RTI | \(u\mapsto[1-u-v+C(u,v)]/(1-u)\) is nondecreasing on \([0,1)\) |
| SI | \(u\mapsto C(u,v)\) is concave |
The SI row is the CDF-section characterization used by the library. Its connection with monotone conditional laws is developed separately.
Two paths from SI to PQD
Transposing the copula reverses the conditioning direction. Total positivity of a CDF, a conditional kernel, and a density are separate predicates; the positive-dependence guide states their proved connections without identifying them by definition.
Library revision: fe53ea2f · Lean 4.34.0