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Unformatted text preview: EXTREME VALUE THEORY Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC 275993260 rls@email.unc.edu AMS Committee on Probability and Statistics Short Course on Statistics of Extreme Events Phoenix, January 11, 2009 1 (From a presentation by Myles Allen) 2 3 4 OUTLINE OF TALK I. Extreme value theory Probability Models Estimation Diagnostics II. Example: North Atlantic Storms III. Example: European Heatwave IV. Example: Trends in Extreme Rainfall Events 5 I. EXTREME VALUE THEORY 6 EXTREME VALUE DISTRIBUTIONS Suppose X 1 ,X 2 ,..., are independent random variables with the same probability distribution, and let M n = max( X 1 ,...,X n ). Un der certain circumstances, it can be shown that there exist nor malizing constants a n > ,b n such that Pr M n b n a n x = F ( a n x + b n ) n H ( x ) . The Three Types Theorem (FisherTippett, Gnedenko) asserts that if nondegenerate H exists, it must be one of three types: H ( x ) = exp( e x ) , all x (Gumbel) H ( x ) = x < exp( x ) x > (Fr echet) H ( x ) = exp( x  ) x < 1 x > (Weibull) In Fr echet and Weibull, > 0. 7 The three types may be combined into a single generalized ex treme value (GEV) distribution: H ( x ) = exp  1 + x ! 1 / + , ( y + = max( y, 0)) where is a location parameter, > 0 is a scale parameter and is a shape parameter. 0 corresponds to the Gumbel distribution, > 0 to the Fr echet distribution with = 1 / , < to the Weibull distribution with = 1 / . > 0: longtailed case, 1 F ( x ) x 1 / , = 0: exponential tail < 0: shorttailed case, finite endpoint at  / 8 EXCEEDANCES OVER THRESHOLDS Consider the distribution of X conditionally on exceeding some high threshold u : F u ( y ) = F ( u + y ) F ( u ) 1 F ( u ) . As u F = sup { x : F ( x ) < 1 } , often find a limit F u ( y ) G ( y ; u , ) where G is generalized Pareto distribution (GPD) G ( y ; , ) = 1 1 + y  1 / + . 9 The Generalized Pareto Distribution G ( y ; , ) = 1 1 + y  1 / + . > 0: longtailed (equivalent to usual Pareto distribution), tail like x 1 / , = 0: take limit as 0 to get G ( y ; , 0) = 1 exp y , i.e. exponential distribution with mean , < 0: finite upper endpoint at / . 10 The PoissonGPD model combines the GPD for the excesses over the threshold with a Poisson distribtion for the number of exceedances. Usually the mean of the Poisson distribution is taken to be per unit time. 11 POINT PROCESS APPROACH Homogeneous case: Exceedance y > u at time t has probability 1 1 + y !...
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 Summer '08
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