Extreme Value Theory: Part II

Sample (N=1000) from a Normal Distribution N(0,1) and fitted curve

Two main kind of Models: Block-maxima and Threshold approaches

Parameter Estimation for GEV Block Maxima

Parameter estimates:

Let:

Maximum likelihood method:

Then:

GEV: Graphical Model Checks

• Probability plot:

• Quantile plots:

• Return level plot:

Return level estimation for high quantiles:

• Profile log-likelihood function:

Example Malawi (Chitedze)

Daily rainfall Seasonal Maxima

MLE parameter estimates

Profile log-likelihood for 100 year return level:

Estimators include:

• Pickands estimator

• Hill‘s estimator

• Deckers-Einmahl-de Haan estimator

Parameter Estimation for GEV Block Maxima

Parameter Estimation II for GEV:

Threshold Models:

Asymptotic model approach:

Threshold models:

Threshold models: Generalized Pareto Distribution

Threshold models: Generalized Pareto Distribution

Threshold selection

Mean residual life plot

Parameter Estimation: Maximum likelihood

Statistical Software for EVT Analysis: Easiest to use

Matlab: EVIM: A Software Package for Extreme Value Analysis by Gencay, Selcuk and Uluguelyagci

Manual and Software: http://www.sfu.ca/~rgencay/evim.html

S-Plus and R: ismev: An Introduction to Statistical Modeling of Extreme Values

Manual and Software: http://cran.r-project.org/web/packages/ismev/index.html

Other software products can be found for example at:

http://www.ral.ucar.edu/~ericg/softextreme.php

http://www.isse.ucar.edu/extremevalues/extreme.html