Extremes of Stationary Time Series [chapter]

2005 Statistics of Extremes  
We begin in section 10.2 by considering the sample maximum, which can be modelled, as for independent sequences, with the generalized extreme value (GEV) distribution. In section 10.3, we achieve a characterization for all exceedances over a high threshold, which supplies a point-process model for clusters of extremes. Models for the extremes of Markov processes are established in section 10.4. Up to this point, we shall only deal with univariate sequences, for which both theory and methods are
more » ... well developed. In section 10.5, we summarize some key results for the extremes of multivariate processes. Finally, in section 10.6, we provide the reader with some key references about additional topics that, despite their importance, did not make it to the core of the chapter. Many of the statistical methods are illustrated for a series of daily maximum temperatures recorded at Uccle, Belgium; see also section 1.3.2. The EXTREMES OF STATIONARY TIME SERIES 371 data analysis was performed using R (version 1.6.1), freely available from www.r-project.org/. The routines for performing the computations in this chapter were written by Chris Ferro and most of them are being incorporated into the R package 'evd' written by Alec Stephenson, freely available from cran.r-project.org. The Sample Maximum Let X 1 , X 2 , . . . be a (strictly) stationary sequence of random variables with marginal distribution function F . The assumption entails that for integer h ≥ 0 and n ≥ 1, the distribution of the random vector (X h+1 , . . . , X h+n ) does not depend on h. For the maximum M n = max i=1,...,n X i , we seek the limiting distribution of (M n − b n )/a n for some choice of normalizing constants a n > 0 and b n . In Chapter 2, it was shown that for independent random variables, the only possible non-degenerate limits are the extreme value distributions. We shall see in section 10.2.1 that this remains true for stationary sequences if long-range dependence at extreme levels is suitably restricted. However, the limit distribution need not be the same as for the maximumM n = max i=1,...,nXi of the associated, independent sequence {X i } with the same marginal distribution as {X i }. The distinction is due to the extremal index, introduced in section 10.2.3, which measures the tendency of extreme values to occur in clusters. The extremal limit theorem For a set J of positive integers, let M(J ) = max i∈J X i . For convenience, also set M(∅) = −∞. We shall partition the integers {1, . . . , n} into disjoint blocks J j = J j,n and show that the block maxima M(J j ) are asymptotically independent. Since M n = max j M(J j ), it follows as in Chapter 2 that the limit distribution of (M n − b n )/a n , if it exists, must be an extreme value distribution. Let (r n ) n be a sequence of positive integers such that r n = o(n) as n → ∞, and put k n = n/r n . Partition {1, . . . , n} into k n blocks of size r n , J j = J j,n = {(j − 1)r n + 1, . . . , jr n }, j = 1, . . . , k n , and, in case k n r n < n, a remainder block, J k n +1 = {k n r n + 1, . . . , n}. Now define thresholds u n increasing at a rate for which the expected number of exceedances over u n remains bounded: lim sup nF (u n ) < ∞, with of courseF = 1 − F . We shall see that, under an appropriate condition, P [M n ≤ u n ] = k n j =1 P [M(J j ) ≤ u n ] + o(1) = (P [M r n ≤ u n ]) k n + o(1), n → ∞. (10.3) This is precisely the desired representation of M n in terms of independent random variables, M r n .
doi:10.1002/0470012382.ch10 fatcat:ef7vabpzhnfcfc2dsp2vpvrzcm