Stochastic Orderings Among Multivariate Extremes

04.10.2024 11:15 – 12:15

RESEARCH INSTITUTE FOR STATISTICS AND INFORMATION SCIENCE: STATISTICS SEMINAR

ABSTRACT

Research on stochastic orderings and inequalities cover several decades. Such orderings will often play a role for robust inference, when only partial knowledge about a highly complex stochastic model is available. Recently, stochastic orderings have also been revisited in the context of isotonic distributional regression. In this work we consider 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>=3 these 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.

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