Source: Ann. Appl. Stat. Volume 4, Number 1
(2010), 203-221.
Classical peaks over threshold analysis is widely used for
statistical modeling of sample extremes, and can be supplemented
by a model for the sizes of clusters of exceedances. Under mild
conditions a compound Poisson process model allows the
estimation of the marginal distribution of threshold exceedances
and of the mean cluster size, but requires the choice of a
threshold and of a run parameter, K, that determines how
exceedances are declustered. We extend a class of estimators of
the reciprocal mean cluster size, known as the extremal index,
establish consistency and asymptotic normality, and use the
compound Poisson process to derive misspecification tests of
model validity and of the choice of run parameter and threshold.
Simulated examples and real data on temperatures and rainfall
illustrate the ideas, both for estimating the extremal index in
nonstandard situations and for assessing the validity of
extremal models.
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