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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 α ) x > (Fr´ echet) H ( x ) = exp( x  α ) x < 1 x > (Weibull) In Fr´ echet and Weibull, α > 0. 7 The three types may be combined into a single generalized ex treme value (GEV) distribution: H ( x ) = exp  1 + ξ x μ ψ ! 1 /ξ + , ( y + = max( y, 0)) where μ is a location parameter, ψ > 0 is a scale parameter and ξ is a shape parameter. ξ → 0 corresponds to the Gumbel distribution, ξ > 0 to the Fr´ echet distribution with α = 1 /ξ , ξ < to the Weibull distribution with α = 1 /ξ . ξ > 0: “longtailed” case, 1 F ( x ) ∝ x 1 /ξ , ξ = 0: “exponential tail” ξ < 0: “shorttailed” case, finite endpoint at μ ξ/ψ 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 → ω F = sup { x : F ( x ) < 1 } , often find a limit F u ( y ) ≈ G ( y ; σ u ,ξ ) where G is generalized Pareto distribution (GPD) G ( y ; σ,ξ ) = 1 1 + ξ y σ 1 /ξ + . 9 The Generalized Pareto Distribution G ( y ; σ,ξ ) = 1 1 + ξ y σ 1 /ξ + . ξ > 0: longtailed (equivalent to usual Pareto distribution), tail like x 1 /ξ , ξ = 0: take limit as ξ → 0 to get G ( y ; σ, 0) = 1 exp y σ , i.e. exponential distribution with mean σ , ξ < 0: finite upper endpoint at σ/ξ . 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 λ per unit time. 11 POINT PROCESS APPROACH Homogeneous case: Exceedance y > u at time t has probability 1 ψ 1 + ξ y μ ψ !...
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This note was uploaded on 02/15/2012 for the course GEO 6938 taught by Professor Staff during the Summer '08 term at University of Florida.
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