Abstract
Among bivariate tail dependence measures, the tail dependence coefficient has emerged as the popular choice. Akin to the correlation matrix, a multivariate dependence measure is constructed using these bivariate measures, and this is referred to in the literature as the tail dependence matrix (TDM). While the problem of determining whether a given d × d matrix is a correlation matrix is of the order O(d3) in complexity, determining if a matrix is a TDM (the realization problem) is believed to be non-polynomial in complexity. Using a linear programming (LP) formulation, we show that the combinatorial structure of the constraints is related to the intractable max-cut problem in a weighted graph. This connection provides an avenue for constructing parametric classes admitting a polynomial in d algorithm for determining membership in its constraint polytope. The complexity of the general realization problem is justifiably of much theoretical interest. Since in practice one resorts to lower dimensional parametrization of the TDMs, we posit that it is rather the complexity of the realization problem restricted to parametric classes of TDMs, and algorithms for it, that are more practically relevant. In this paper, we show how the inherent symmetry and sparsity in a parametrization can be exploited to achieve a significant reduction in the LP formulation, which can lead to polynomial complexity of such realization problems - some parametrizations even resulting in the constraint polytope being independent of d. We also explore the use of a probabilistic viewpoint on TDMs to derive the constraint polytopes.
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Acknowledgements
Our interest in the realization problem arose from discussions with Paul Embrechts, to whom we owe a debt of gratitude. We thank an associate editor for bringing our attention to Krause et al. (2017), and an area editor and two referees for many thoughtful suggestions. We thank Jonas Schwinn and Ralf Werner for graciously sharing their well executed implementation of the KSSW algorithm without which parts of this work would not have been possible. Also, we would like to thank Sam Burer and Ruodu Wang for fruitful discussions. The first author would like to acknowledge with gratitude the support from a Society of Actuaries’ Center of Actuarial Excellence Research Grant.
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A Appendix
A Appendix
A Matrix and its Associated SageMath Code to Compute N(B)
The Constraint Polytope when (d1, d2) = (2, 2)
The Constraint Polytope when (d1, d2) = (3, 3)
The Constraint Polytope when (d1, d2) = (2, 4)
The Constraint Polytope when (d1, d2) = (4, 4)
Construction of Sets Ai, i = 1, … , d, for any d: 2-Dependent TDM of Proposition 21
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Shyamalkumar, N.D., Tao, S. On tail dependence matrices. Extremes 23, 245–285 (2020). https://doi.org/10.1007/s10687-019-00366-y
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DOI: https://doi.org/10.1007/s10687-019-00366-y