Extreme Value Distributions AMS2009

Extreme Value Distributions AMS2009 - EXTREME VALUE THEORY...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EXTREME VALUE THEORY Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC 27599-3260 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 (Fisher-Tippett, 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: long-tailed case, 1- F ( x ) x- 1 / , = 0: exponential tail < 0: short-tailed 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: long-tailed (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 Poisson-GPD 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- !...
View Full Document

Page1 / 58

Extreme Value Distributions AMS2009 - EXTREME VALUE THEORY...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online