Stochastic ordering in multivariate extremes

Abstract

The article considers the multivariate stochastic orders of upper orthants, lower orthants and positive quadrant dependence (PQD) among simple max-stable distributions and their exponent measures. It is shown for each order that it holds for the max-stable distribution if and only if it holds for the corresponding exponent measure. The finding is non-trivial for upper orthants (and hence PQD order). From dimension $d\geq 3$ these three orders are not equivalent and a variety of phenomena can occur. However, every simple max-stable distribution PQD-dominates the corresponding independent model and is PQD-dominated by the fully dependent model. Among parametric models the asymmetric Dirichlet family and the Hüsler-Reiss family turn out to be PQD-ordered according to the natural order within their parameter spaces. For the Hüsler-Reiss family this holds true even for the supermodular order.

Figures (8)

• Figure 1: Angular densities (heat maps) of the symmetric max-stable Dirichlet model (top) and of the asymmetric max-stable Dirichlet model (bottom), cf. \ref{['eq:DirichletAngularDensity']} for an expression of the density. Larger values are represented by brighter colours. The corresponding max-stable distributions are stochastically ordered in the PQD sense, increasing from left to right (Theorem \ref{['thm:Dirichlet-PQD']}). The black, blue and red boxes encode the matching with Figures \ref{['fig:MinCDFs']} and \ref{['fig:MaxCDFs']}.
• Figure 2: Nested max-zonoids and Pickands dependence functions ranging from full dependence (black), an asymmetric Dirichlet model with ${\bm \alpha}=(30,0.2)$ (dark grey), its associated Choquet (Tawn-Molchanov) model (light grey) to the fully independent model (white).
• Figure 3: Illustration of test sets for multivariate orders for exponent measures in dimension $d=2$, cf. Definition \ref{['def:orders-Lambda']}. Left: $\Lambda$ is locally finite on the (closed) grey area for all $\varepsilon>0$, its total (infinite) mass is contained in the union of such sets; middle: admissible complement of a lower orthant $\mathbb{R}^2\setminus L_{\bm a}$ (blue area) for testing lower orthant order for $\Lambda$; right: admissible upper orthants $U_{\bm a}$, $U_{\bm b}$, $U_{\bm c}$ (three red areas) for testing upper orthant order for $\Lambda$.
• Figure 4: Top: Nested max-zonoids (left) and ordered (hypographs of) Pickands dependence functions (right) from the fully symmetric Dirichlet family for $\alpha \in \lbrace 0.0625, 0.25, 1, 4\rbrace$. Smaller values of $\alpha$ correspond to larger sets and larger Pickands dependence functions and are closer to the independence model represented by the box $[0,1]^2$ or the constant function, which is identically 1. The fully dependent model is represented in black. Bottom: Non-nested max-zonoids and non-ordered Pickands dependence function from the asymmetric Dirichlet family for $(\alpha_1,\alpha_2) \in \{(0.15,12),(4,0.2)\}$.
• Figure 5: Nested max-zonoids and ordered Pickands dependence functions of the asymmetric max-stable Dirichlet family for $(\alpha_1,\alpha_2)\in\lbrace(0.25,0.25),(1,0.25),(1,1),(1,4),(4,4)\rbrace$. Componentwise smaller values of $(\alpha_1,\alpha_2)$ correspond to larger sets and larger Pickands dependence functions and are closer to the independence model.

Theorems & Definitions (48)

• Lemma 2.2
• Lemma 2.4: Closure of Dirichlet model under taking marginals
• Lemma 2.6: Closure of Hüsler-Reiß model under taking marginals
• Theorem 2.7: Triangular array convergence of maxima of Gaussian vectors, cf. kabluchko_11 Theorem 2
• Remark 2.8
• Theorem 2.9
• Example 2.10: Choquet model in the bivariate case
• Definition 3.1: Multivariate orders LO, UO, PQD, shsh_07, Sections 6.G and 9.A, mueller_02, Sections 3.3. and 3.8
• Definition 3.2: Multivariate orders for exponent measures
• Remark 3.3