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Extreme Value Theory: Part II

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. Maximum likelihood method:. Let:. Then:. Parameter estimates:.

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Extreme Value Theory: Part II

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  1. Extreme Value Theory: Part II Sample (N=1000) from a Normal Distribution N(0,1) and fitted curve

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

  3. Parameter Estimation for GEV Block Maxima Maximum likelihood method: Let: Then: Parameter estimates:

  4. Return level estimation for high quantiles: GEV: Graphical Model Checks • Quantile plots: • Probability plot: • Return level plot: • Profile log-likelihood function:

  5. Example Malawi (Chitedze) Daily rainfall Seasonal Maxima MLE parameter estimates

  6. Profile log-likelihood for 100 year return level:

  7. Parameter Estimation for GEV Block Maxima • Estimators include: • Pickands estimator • Hill‘s estimator • Deckers-Einmahl-de Haan estimator

  8. Parameter Estimation II for GEV:

  9. Threshold Models:

  10. Asymptotic model approach:

  11. Threshold models:

  12. Threshold models: Generalized Pareto Distribution

  13. Threshold models: Generalized Pareto Distribution

  14. Threshold selection Mean residual life plot

  15. Parameter Estimation: Maximum likelihood

  16. 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

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