bibkey: puccettiwang2015dependence authors: Giovanni Puccetti and Ruodu Wang year: 2015 title: Extremal Dependence Concepts doi: 10.1214/15-STS525 url: https://arxiv.org/abs/1512.03232v3 claim: “Bivariate distributions with fixed marginals have sharp lower and upper Frechet-Hoeffding bounds, attained by countermonotonic and comonotonic couplings. In any finite dimension, comonotonic couplings maximize mean-compatible supermodular expectations within the fixed-marginal class.” strata_touched:

• D5/S3/Estimation/DataProcessing/FiniteSimplexFiberPolytope
• D5/S3/Combinatorics/Probability/FiniteCommonMarginalSupermodular license: citation-only triage: anchor

Fixed marginals, sharp couplings and target dependence

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DOI: 10.1214/15-STS525. Author text: https://arxiv.org/abs/1512.03232v3. Section 2, Theorem 2.1(d), equation (2.3), states that a comonotonic coupling maximizes the expectation of every mean-compatible supermodular function over all joint laws with the prescribed one-dimensional marginals. The bivariate bounds used below are in Theorem 2.1(b), Theorem 3.1(b), equations (3.2) and (3.5).

Finite common-support comparison

For four equal marginals on a nonempty finite linearly ordered support, the comonotonic law is the diagonal law: a scalar value with marginal mass m(t) is placed at (t,t,t,t). Theorem 2.1(d) therefore gives the finite common-marginal supermodular expectation comparison.

An order-preserving embedding of the finite support into the real line preserves coordinatewise minimum and maximum. A monotone step map back to that support, fixing each embedded support value and clipped at the two endpoints, preserves these operations as well. Composing a lattice-supermodular test with this map in each coordinate extends it to a bounded supermodular real test. Thus mean compatibility is automatic on the finite support, including for signed tests. The joint and marginal masses may be arbitrary real probabilities; zero masses, repeated coordinates and singleton supports are included.

This identifies the classical comparison represented by the finite Lean theorem. It supplies no continuum limiting passage or supermodularity proof for a particular test chosen in an application.

Bivariate coupling bounds

Giovanni Puccetti and Ruodu Wang, Extremal Dependence Concepts, Statistical Science 30(4), 485–517. The directly consumed author text is arXiv:1512.03232v3.

Section 1 distinguishes the joint distribution from its one-dimensional marginals and defines the Fréchet class. Section 2, Theorem 2.1(b), gives the comonotonic joint distribution $F_2^{\vee }(u,v)=\min (F_1(u),F_2(v))$. Section 3.1, Theorem 3.1(b), equation (3.2), gives the countermonotonic distribution $F_2^{\wedge }(u,v)=\max (F_1(u)+F_2(v)-1,0)$. Equation (3.5) states the sharp bivariate ordering between them. The original author PDF’s pp. 11–12 contain these lower-bound clauses. These are classical bounds reviewed by the authors, not results newly attributed to this repository or priority claims about their discovery.

For two Bernoulli events $A,B$ with probabilities $u,v$, let $k=P(A\cap B)$. Their four-cell law is

$$(p_{00},p_{10},p_{01},p_{11})=(1-u-v+k,u-k,v-k,k).$$

Nonnegativity is equivalent to $\max (0,u+v-1)\le k\le \min (u,v)$; each endpoint is attainable. This is the finite-event application of the bivariate bounds. If an additional disjoint mode has probability $Z$, condition on its complement of mass $r=1-Z>0$ and substitute $u=X/r$, $v=Y/r$, $k=\kappa /r$. Multiplication by $r$ gives

$$\max (0,X+Y+Z-1)\le \kappa \le \min (X,Y).$$

At $r=0$, the extra mode has all probability and $X=Y=\kappa =0$; conditioning is unnecessary. The bounds thus apply to the entire five-mode domain, including its apex.

For fair Bernoulli marginals, the laws $\frac{1}{2}({\delta }_{00}+{\delta }_{11})$ and $\frac{1}{2}({\delta }_{01}+{\delta }_{10})$ have the same marginal data. An additive target has the same expectation under both, while the distribution of that target need not agree. This is a classical coupling input. Its realization by actual legal native FIB windows and its continuing-reply consequences are supplied by the occupancy-pyramid volume, not by this probability paper. No theorem about a native FIB reader, quantum instrument or acquired archive is attributed to the paper.