FGEV2 x0ut [FGEV1 x0ut0621

where 0 <0 < 1. The parameter 0 linking the dependence and independence cases is called extremal index.

Equation (6.21) has considerable practical consequences because it allows to apply the GEV and GP estimation methods also to data from short-memory processes. A caveat here is that the number of indepen dent observations is reduced. Coles (2001b: Section 5.3.1 therein) gives the number of n0 as effective data size with respect to the quality of the GEV approximation.

For fitting a GP distribution to threshold extremes from short-memory processes, Coles (2001b: Section 5.3.3 therein) notes the technique of declustering. This takes into account that under persistence the extremes tend to occur in clusters (consecutive times). Within a cluster, only the maximum excess over a threshold is retained for GP estimation. Declustering is equivalent to POT data selection (Section 6.1.2) with an imposed secondary selection criterion. To prohibit the information loss associated with declustering, it may be worth instead to consider to retain all POT values and account for the persistence by either modelling it or adjusting the covariance matrix (see background material).

6.2.5.3 Long memory

Even if X(i) is a long-memory process (Section 2.4.1), the GEV or GP model may be applicable. Smith (1989: pp. 392-393 therein) remarks that if X(i) has a Gaussian distributional shape and p(n) log(n) ^ 0 for n ^ to, then the long-range dependence "does not matter," referring to a paper by Berman (1964). The autocorrelation function p(h) for an ARFIMA process does indeed fulfill the condition, and a suitable transformation of X(i) may yield approximately a Gaussian shape. See the example of river runoff (Section 2.5.3).

The major problem from long memory could be that the number of independent observations is reduced—to a stronger degree than for short memory. This makes the GEV or GP approximation less accurate than in the no-memory or short-memory cases.

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