Abstract
The tail correlation function (TCF) is a popular bivariate extremal dependence measure to summarize data in the domain of attraction of a max-stable process. For the class of TCFs, being largely unexplored so far, several aspects are contributed: (i) generalization of some mixing max-stable processes (ii) transfer of two geostatistical construction principles to max-stable processes, including the turning bands operator (iii) identification of subclasses of TCFs, including M3 processes based on radial monotone shapes (iv) recovery of subclasses of max-stable processes from TCFs (v) parametric classes (iv) diversity of max-stable processes sharing an identical TCF. We conclude that caution should be exercised when using TCFs for statistical inference.
Highlights
The tail correlation function (TCF) χ of a stationary process X on Rd is defined through the following limit provided that it exists χ (t )
Parametric subclasses of max-stable processes have been fitted to environmental spatial data and the extremal coefficient function (ECF) θ is usually considered in order to assess the goodness of fit (Blanchet and Davison 2011; Engelke et al 2012b; Davison and Gholamrezaee 2012; Davison et al 2012; Schlather and Tawn 2003; Thibaud and Opitz 2014)
The stationary truncation generalizes a construction described in Schlather (2002, p. 39) and corresponds to the multiplication of a given TCF with another TCF that has compact support. It can shorten the range of tail dependence, e.g. to a compact set, a feature which is of interest for modelling purposes, cf
Summary
The tail correlation function (TCF) χ of a stationary process X on Rd is defined through the following limit provided that it exists χ (t ). Parametric subclasses of max-stable processes have been fitted to environmental spatial data and the ECF θ (that is equivalent to the TCF χ) is usually considered in order to assess the goodness of fit (Blanchet and Davison 2011; Engelke et al 2012b; Davison and Gholamrezaee 2012; Davison et al 2012; Schlather and Tawn 2003; Thibaud and Opitz 2014) All these references contain plots comparing non-parametric estimates of extremal coefficients to the theoretical ECFs. While continuous correlation functions can be characterized by means of Bochner’s theorem as Fourier transforms of probability measures, no such characterization is available for the subclass of (continuous) TCFs. At least, Fiebig et al (2014) show that the set of TCFs on an arbitrary space T is closed under convex combinations, products and pointwise limits and provide necessary conditions for a function to be a tail correlation function. An integral of the form f (x) dF (x), where F is a monotone function, is always meant in the Riemann-Stieltjes sense
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
