Speaker: Dan Cooley (Colorado State)
Title: Regular Variation, the TPDM, and Vector Spaces
Abstract:
In high dimensions, fully characterizing the extremal dependence structure is difficult. However, summarizing extremal dependence via pairwise measures, whose estimation is straightforward, can provide actionable dependence information for modeling. Assuming the framework of multivariate regular variation, this talk will introduce the tail pairwise dependence matrix (TPDM), which can be viewed as an extremal analogue to the covariance matrix. The talk will briefly review how the TPDM has been used to perform extremal PCA, to model extremal time series, and to define partial tail correlation.
The talk will then turn attention toward connecting standard ideas about vector spaces and inner products to regular variation and the TPDM. One challenge is that collections of regularly varying random variables cannot be assumed to be jointly regularly varying. In ongoing work, we define a vector space of regularly varying random variables, and assign a norm that is related to the `scale’ of the random variable’s tail. Current work is to show that the TPDM corresponds to an inner product in the case when the tail index alpha = 2.
This is joint work with Kenneth Broadhead, CSU.
