Paper: arXiv 2604.23983

Authors: Janusz Milek

Abstract

We study multivariate tail-dependence compatibility for complete and partial signed tail families, treating lower-tail, upper-tail, and mixed configurations in one geometric witness representation indexed by active coordinate sets and sign patterns. For a complete signed tail family, witness generator weights w = (w_{I,sigma}) give a linear incidence parametrization and are recovered by explicit triangular inversion. Excluding the geometric scale p0, the complete case uses 3^d - 1 generator weights, matching the number of complete signed tail coefficients; for partial specifications, only selected target coefficients need be prescribed. At a fixed threshold p0 in (0, 1/2), the inversion identifies the normalized noncentral ternary cell masses of any realizing copula. Hence finite-threshold compatibility is characterized by nonnegative recovered generator weights, singleton normalization, and the residual central-mass constraint. This yields a complete Moebius-type synthesis within the witness framework. If the recovered increments are nonnegative and singleton normalization holds, then S(w) = sum(w) determines the admissible finite-scale range, and every admissible p0 gives an exact witness realization. In the canonical ray geometry, such a realization preserves the same complete signed tail family throughout 0 < p <= p0. Thus the primary object is the complete signed tail family lambda: it is realized at every admissible finite scale and can be carried along families of witness copulas with p0 decreasing to 0. Partial, noisy, or inconsistent specifications are treated through linear-feasibility and weighted-l1 recovery problems in the same parametrization. The representation separates the p0-free incidence/Moebius layer from finite-threshold realization and provides tools for realization, simulation, calibration, completion, repair, and scenario design.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly novel and mathematically rigorous geometric framework for resolving signed multivariate tail-dependence compatibility using Möbius-type inversion and incidence geometry. However, it is purely theoretical with minimal empirical validation, backtesting, or publicly available code for practical implementation.

Research Flowchart

  flowchart TD
    A["Research Goal: Characterize & Synthesize<br/>Signed Multivariate Tail-Dependence Compatibility"] --> B["Input: Signed Tail Coefficients<br/>(Complete, Partial, or Noisy Specification)"]
    
    B --> C["Geometric Witness Framework:<br/>Map coefficients via Linear Incidence & Triangular Moebius Inversion"]
    
    C --> D{"Feasibility & Optimization Check:<br/>Nonnegative weights w & Singleton/Central-mass constraints"}
    
    D -- "Valid / Admissible" --> E["Exact Synthesis:<br/>Determine scale range p0 & Construct Witness Copula Realizations"]
    D -- "Inconsistent / Partial" --> F["Repair & Calibration:<br/>Solve Linear-Feasibility & Weighted L1 Recovery"]
    
    E --> G["Outcomes: Scale-Invariant Tail Preservation,<br/>Simulation, Completion & Scenario Design"]
    F --> G