\contentsline {chapter}{\numberline {1}Challenges of Applied Extreme Value Statistics}{1}
\contentsline {section}{\numberline {1.1}Introduction}{2}
\contentsline {section}{\numberline {1.2}A Brief Summary of Status, Problems and Challenges}{2}
\contentsline {chapter}{\numberline {2}Classical Extreme Value Theory}{5}
\contentsline {section}{\numberline {2.1}Introduction}{6}
\contentsline {section}{\numberline {2.2}The Asymptotic Limits of Extreme Value Distributions}{6}
\contentsline {section}{\numberline {2.3}The Block Maxima Method}{8}
\contentsline {section}{\numberline {2.4}Outline Proof of the Extremal Types Theorem}{9}
\contentsline {section}{\numberline {2.5}Domains of Attraction for the Extreme Value Distributions}{10}
\contentsline {section}{\numberline {2.6}Parameter Estimation for the GEV Distributions}{11}
\contentsline {subsection}{\numberline {2.6.1}Estimation by the method of moments}{12}
\contentsline {subsection}{\numberline {2.6.2}Maximum likelihood estimation}{12}
\contentsline {section}{\numberline {2.7}Model Validation}{13}
\contentsline {section}{\numberline {2.8}Estimating Confidence Intervals by Bootstrapping}{14}
\contentsline {section}{\numberline {2.9}The Asymptotic Extreme Value Distributions for Dependent Sequences}{15}
\contentsline {chapter}{\numberline {3}The Peaks-Over-Threshold Method}{19}
\contentsline {section}{\numberline {3.1}Introduction}{20}
\contentsline {section}{\numberline {3.2}The Peaks-Over-Threshold Method}{20}
\contentsline {section}{\numberline {3.3}Threshold Selection}{22}
\contentsline {section}{\numberline {3.4}Return Periods}{23}
\contentsline {section}{\numberline {3.5}Parameter Estimation for the GP Distributions}{24}
\contentsline {subsection}{\numberline {3.5.1}de Haan estimators}{24}
\contentsline {subsection}{\numberline {3.5.2}Moment estimators}{25}
\contentsline {subsection}{\numberline {3.5.3}Maximum likelihood estimators}{25}
\contentsline {section}{\numberline {3.6}Model Validation}{26}
\contentsline {section}{\numberline {3.7}Estimating Confidence Intervals by Bootstrapping}{26}
\contentsline {chapter}{\numberline {4}A Point Process Approach to Extreme Value Statistics}{29}
\contentsline {section}{\numberline {4.1}Introduction}{30}
\contentsline {section}{\numberline {4.2}Average Rate of Level Crossings}{30}
\contentsline {section}{\numberline {4.3}Distribution of Peaks of a Narrow-banded Process}{33}
\contentsline {section}{\numberline {4.4}Average Upcrossing Rate and Distribution of Peaks of a Gaussian Process}{34}
\contentsline {section}{\numberline {4.5}Extreme Value Distributions by the Upcrossing Rate Method}{37}
\contentsline {section}{\numberline {4.6}Extreme Values of Gaussian Processes}{39}
\contentsline {section}{\numberline {4.7}The Crossing Rate of Transformed Processes}{42}
\contentsline {section}{\numberline {4.8}Hermite Moment Models}{43}
\contentsline {section}{\numberline {4.9}Return Period}{44}
\contentsline {section}{\numberline {4.10}Long Term Extreme Value Distributions}{45}
\contentsline {subsection}{\numberline {4.10.1}All peak values}{46}
\contentsline {subsection}{\numberline {4.10.2}All short-term extremes}{47}
\contentsline {subsection}{\numberline {4.10.3}The long-term extreme value}{48}
\contentsline {subsection}{\numberline {4.10.4}Simplified methods}{50}
\contentsline {section}{\numberline {4.11}Appendix}{52}
\contentsline {chapter}{\numberline {5}The ACER Method}{55}
\contentsline {section}{\numberline {5.1}Introduction}{56}
\contentsline {section}{\numberline {5.2}A Sequence of Conditioning Approximations}{57}
\contentsline {section}{\numberline {5.3}Empirical Estimation of the Average Conditional Exceedance Rates}{60}
\contentsline {section}{\numberline {5.4}Long-Term Extreme Value Analysis by the ACER Method}{63}
\contentsline {section}{\numberline {5.5}Estimation of Extremes for the Asymptotic Gumbel Case}{64}
\contentsline {section}{\numberline {5.6}Estimation of Extremes for the General Case}{67}
\contentsline {chapter}{\numberline {6}Some Practical Aspects of Extreme Value Analyses}{69}
\contentsline {section}{\numberline {6.1}Introduction}{70}
\contentsline {section}{\numberline {6.2}Extreme Value Prediction for Synthetic Data}{70}
\contentsline {section}{\numberline {6.3}Measured Wind Speed Data}{72}
\contentsline {section}{\numberline {6.4}Extreme Value Prediction for a Narrow Band Process}{81}
\contentsline {chapter}{\numberline {7}Estimation of Extreme Values for Financial Risk Assessment}{87}
\contentsline {section}{\numberline {7.1}Introduction}{88}
\contentsline {section}{\numberline {7.2}Value-at-Risk}{88}
\contentsline {section}{\numberline {7.3}Application to Simulated Time Series of Electricity Prices}{89}
\contentsline {section}{\numberline {7.4}Electricity Price Data}{90}
\contentsline {section}{\numberline {7.5}Conditional Approach}{90}
\contentsline {section}{\numberline {7.6}Results}{92}
\contentsline {subsection}{\numberline {7.6.1}Unconditional approach}{93}
\contentsline {subsection}{\numberline {7.6.2}Conditional approach}{94}
\contentsline {chapter}{\numberline {8}The Upcrossing Rate via the Characteristic Function}{97}
\contentsline {section}{\numberline {8.1}Introduction}{98}
\contentsline {section}{\numberline {8.2}The Response Process}{98}
\contentsline {section}{\numberline {8.3}The Average Crossing Rate }{101}
\contentsline {section}{\numberline {8.4}Numerical Calculation}{102}
\contentsline {section}{\numberline {8.5}Numerical Examples}{105}
\contentsline {subsection}{\numberline {8.5.1}Slow-drift response}{105}
\contentsline {subsection}{\numberline {8.5.2}Moored deep floater}{106}
\contentsline {subsection}{\numberline {8.5.3}Wind excited structure}{110}
\contentsline {section}{\numberline {8.6}Appendix 1 - The Average Crossing Rate}{113}
\contentsline {section}{\numberline {8.7}Appendix 2 - The Characteristic Function}{114}
\contentsline {chapter}{\numberline {9}Monte Carlo Methods and Extreme Value Estimation}{119}
\contentsline {section}{\numberline {9.1}Introduction}{120}
\contentsline {section}{\numberline {9.2}Simulation of Stationary Stochastic Processes}{120}
\contentsline {subsection}{\numberline {9.2.1}Realizations of stochastic processes }{120}
\contentsline {subsection}{\numberline {9.2.2}Variance spectra}{122}
\contentsline {subsection}{\numberline {9.2.3}Units of variance spectra}{124}
\contentsline {subsection}{\numberline {9.2.4}Example - A realization of a wave process}{125}
\contentsline {subsection}{\numberline {9.2.5}The variance spectrum directly from the realizations}{127}
\contentsline {section}{\numberline {9.3}Monte Carlo Simulation of Load and Response}{128}
\contentsline {section}{\numberline {9.4}Sample Statistics of Simulated Response}{129}
\contentsline {section}{\numberline {9.5}Latin Hypercube Sampling}{131}
\contentsline {section}{\numberline {9.6}Estimation of Extreme Response}{131}
\contentsline {subsection}{\numberline {9.6.1}The Gumbel method}{132}
\contentsline {subsection}{\numberline {9.6.2}The point process method}{133}
\contentsline {subsection}{\numberline {9.6.3}A comparison of methods}{135}
\contentsline {subsection}{\numberline {9.6.4}Combination of multiple stochastic load effects}{140}
\contentsline {subsection}{\numberline {9.6.5}Total surge response of a TLP}{148}
\contentsline {chapter}{\numberline {10}Bivariate Extreme Value Distributions}{157}
\contentsline {section}{\numberline {10.1}Introduction}{158}
\contentsline {section}{\numberline {10.2}Componentwise Extremes}{159}
\contentsline {section}{\numberline {10.3}Bivariate ACER Functions}{160}
\contentsline {section}{\numberline {10.4}Functional Representation of the Empirically Estimated Bivariate ACER Functions}{162}
\contentsline {section}{\numberline {10.5}Numerical Examples}{164}
\contentsline {subsection}{\numberline {10.5.1}Wind speed measured at two adjacent weather stations}{164}
\contentsline {subsection}{\numberline {10.5.2}Wind speed and wave height measured at a North Sea weather station}{169}
\contentsline {section}{\numberline {10.6}Appendix 1: The Sequence of Conditioning Approximations}{174}
\contentsline {section}{\numberline {10.7}Appendix 2: Empirical Estimation of the Bivariate ACER Functions}{177}
\contentsline {chapter}{\numberline {11}Space-Time Extremes of Random Fields}{181}
\contentsline {section}{\numberline {11.1}Introduction}{182}
\contentsline {section}{\numberline {11.2}Spatial-Temporal Extremes for Gaussian Random Fields}{183}
\contentsline {section}{\numberline {11.3}A Simplified Approach}{186}
\contentsline {section}{\numberline {11.4}Spatial-Temporal Extremes for Non-Gaussian Random Fields}{188}
\contentsline {section}{\numberline {11.5}Empirical Estimation of the Mean Upcrossing Rate}{190}
\contentsline {section}{\numberline {11.6}Numerical Examples for Gaussian Random Fields}{191}
\contentsline {subsection}{\numberline {11.6.1}1+1-dimensional Gaussian field}{192}
\contentsline {subsection}{\numberline {11.6.2}1+1-dimensional Gaussian sea}{193}
\contentsline {subsection}{\numberline {11.6.3}A short-crested Gaussian sea}{196}
\contentsline {section}{\numberline {11.7}Numerical Examples for Non-Gaussian Random Fields}{199}
\contentsline {subsection}{\numberline {11.7.1}A second-order wave field}{199}
\contentsline {subsection}{\numberline {11.7.2}A Student's $t$ random field}{203}
\contentsline {section}{\numberline {11.8}Comments}{205}
\contentsline {chapter}{\numberline {12}A Case Study - Extreme Water Levels}{209}
\contentsline {section}{\numberline {12.1}Introduction}{210}
\contentsline {section}{\numberline {12.2}Data Sets}{211}
\contentsline {section}{\numberline {12.3}Annual Maxima Method}{212}
\contentsline {subsection}{\numberline {12.3.1} Application to water level measurements}{212}
\contentsline {section}{\numberline {12.4}The Peaks-Over-Threshold Method}{215}
\contentsline {subsection}{\numberline {12.4.1} Application to water level measurements}{217}
\contentsline {section}{\numberline {12.5}Revised Joint Probabilities Method}{223}
\contentsline {subsection}{\numberline {12.5.1} Estimating return levels with the RJP method}{225}
\contentsline {subsection}{\numberline {12.5.2} Application to water level measurements}{226}
\contentsline {section}{\numberline {12.6}The ACER Method}{229}
\contentsline {subsection}{\numberline {12.6.1} Application to water level measurements}{229}
\contentsline {section}{\numberline {12.7}Discussion of Results}{232}
\contentsline {subsection}{\numberline {12.7.1} Oslo}{232}
\contentsline {subsection}{\numberline {12.7.2} Heimsjø}{234}
\contentsline {subsection}{\numberline {12.7.3} Honningsvåg}{234}
\contentsline {subsection}{\numberline {12.7.4} Comments}{234}
\contentsline {chapter}{References}{239}
\contentsline {chapter}{Index}{251}