Total positivity in multivariate extremes

Abstract

Positive dependence is present in many real world data sets and has appealing stochastic properties. In particular, the notion of multivariate total positivity of order 2 (MTP 2 ) is a convex constraint and acts as an implicit regularizer in the Gaussian case. We study positive dependence in multivariate extremes and introduce EMTP 2 , an extremal version of MTP 2 . This notion turns out to appear prominently in extremes and, in fact, it is satisfied by many classical models. For a Hüsler-Reiss distribution, the analogue of a Gaussian distribution in extremes, we show that it is EMTP 2 if and only if its precision matrix is a Laplacian of a connected graph. We propose an estimator for the parameters of the Hüsler-Reiss distribution under EMTP 2 as the solution of a convex optimization problem with Laplacian constraint. We prove that this estimator is consistent and typically yields a sparse model with possibly non-decomposable extremal graphical structure. At the example of two data sets, we illustrate this regularization and the superior performance compared to existing methods.

Optimal designs for discrete choice models via graph Laplacians

“…6.2). In fact, we show that for the Bradley-Terry paired comparison model (k = 2), the linear inequality constraints in Proposition 7 allow a reformulation so that the design problem is directly equivalent to Proposition 6.2 in Röttger et al (2021), see Corollaries 11 and 12. The optimization problem (1) also has appealing properties for high-dimensional settings, as the MLE of Laplacian-constrained Gaussian graphical models exists for sample size 1 (Ying et al, 2021).…”

“…For an extensive treatment, see e.g. Devriendt (2022) or Appendix A in Röttger et al (2021). Let G = (V, E) be a simple undirected graph with vertex set V = {1, .…”

“…The matrix Γ lies in S m 0 , the set of symmetric m × m matrices with zero diagonal. Furthermore, the matrix A is positive definite if an only if Γ ∈ C m , where C m ⊂ S m 0 is the cone of conditionally negative definite symmetric m × m matrices Röttger et al, 2021). The inverse Farris transform reconstructs the entries of A via…”

“…This is equivalent to the dual optimization problem for Gaussian maximum likelihood estimation under Laplacian constraints, compare for example Röttger et al (2021) or Ying et al (2021). In fact, Corollary 11 equals Proposition 6.2 of Röttger et al (2021) with Γ = (m − 1)L −1 R1 in matrix form.…”

“…These models are an active research area in machine learning, see e.g. Egilmez et al (2017), Kumar et al (2019) or Ying et al (2021), and multivariate extremes (Röttger et al, 2021). Laplacian matrices are very relevant in combinatorics and graph theory and connect geometric, graphical and algebraic properties, see for example Devriendt (2022) for a concise introduction.…”