Extremes and Local Dependence in Stationary Sequences.

Abstract

Oct EX71MHES AND LOCAL DEPENDENCE IN DISTRIBUTION STATEMENT (of this Report)Approved for public release; distribution unlimited. Extremes; maxima; stationary processes. ABSTRACT (Continue an reverse side if necessary and identify by block numnber)7Extensions of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction: (a) a weak 'mixing condition' restrictine long range dependence; (b) a local condition restricting the 'clustering' of high level exceedances. The purpose of this paper is to investigate extremal properties when the local condition (b) is omitted. It is found that, under general conditions, the type of the limiting distribution for maxima is unaltered. The precise modifications and degree of clustering of . high level exceedances are found to be largely described by a-4C4N4tPJEM) DID JA t 1473 EDITION OF I NOV 65IS 08SOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (Wh~en Date Entered) Vp4. The purpose of this paper is to investigate extremal properties when the local condition (b) is omitted. It is found that, under general conditions, the type of the limiting distribution for maxima is unaltered. The precise modifications and the degree of clustering of high level exceedances are found to be largely described by a parameter here called the "extremal index" of the sequence. SECUftIT1~ CLASSIFICATION OF TIlS A61(WMan

Examining Flood Frequency Distributions in the Midwest U.S.1

“…For hydrological applications, the GEV distribution has been widely used to parametrically describe the flood records (e.g., Stedinger and Lu, 1995;Hosking and Wallis, 1997;Katz et al, 2002). From a theoretical standpoint, the GEV distribution represents the limiting distribution of a series of maxima of independent (or weakly dependent) and identically distributed random variables (e.g., Leadbetter, 1983). The GEV distribution can also be described in terms of seasonal mixtures of exponentially or GEV distributed random variables (e.g., Waylen and Woo, 1982;Rossi et al, 1984;Morrison and Smith, 2002;Villarini and Smith, 2010).…”

Generalized Additive Modelling of Sample Extremes

“…Problem (b) could be solved by identifying independent clusters, retaining only their maxima, and fitting the Poisson process model to these maxima. A theoretical basis for this is given by Leadbetter (1983) and Anderson (1990), but there is a possible loss of information and a declustering algorithm is needed. An alternative is to construct a detailed model for intracluster behaviour.…”

Estimating Probabilities of Extreme Sea-Levels

“…To improve the crude empirical distribution function estimator for hourly surges, used by Pugh and Vassie (1979), we need to smooth and extrapolate the tail of this distribution and require some theoretical basis to justify the form of the extrapolation. The theory of extreme values for stationary sequences contains relevant ideas (O'Brien, 1974;Leadbetter, 1983). In particular, provided that a weak mixing condition holds (see Leadbetter et al (1983), chapter 3), then for a stationary sequence s., ...,~n Here the parameter fJ, 0 :s:; fJ :s:; 1, called the extremal index, is defined as follows: let the cluster size for each independent extreme event, which exceeds level z, be the number of observations above z during that event.…”