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Advisor(s)
Abstract(s)
The asymptotic results that underlie applications of extreme random fields
often assume that the variables are located on a regular discrete grid, identified
with Z2, and that they satisfy stationarity and isotropy conditions. Here we extend
the existing theory, concerning the asymptotic behavior of the maximum and the
extremal index, to non-stationary and anisotropic random fields, defined over discrete
subsets of R2.We show that, under a suitable coordinatewise mixing condition,
the maximum may be regarded as the maximum of an approximately independent
sequence of submaxima, although there may be high local dependence leading to
clustering of high values. Under restrictions on the local path behavior of high values,
criteria are given for the existence and value of the spatial extremal index which
plays a key role in determining the cluster sizes and quantifying the strength of dependence
between exceedances of high levels. The general theory is applied to the class
of max-stable random fields, for which the extremal index is obtained as a function
of well-known tail dependence measures found in the literature, leading to a simple
estimation method for this parameter. The results are illustrated with non-stationary
Gaussian and 1-dependent random fields. For the latter, a simulation and estimation
study is performed.
Description
Keywords
Random field Max-stable process Extremal dependence Spatial extremal index