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Title:
Extremes in nature : an approach using copulas
Publication Information:
Dordrecht, The Netherlands : Springer, 2007
ISBN:
9781402044144
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Available online version
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30000010150253 GB5014 E87 2007 Open Access Book Book
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Summary

Summary

The most powerful earthquake in 40 years occurred on 26th December 2004 off the west coast of Sumatra, Indonesia. The tsunami it generated turned into one of the worst known natural disasters when walls of water crashed across the Indian Ocean, causing waves to reach Somalia in Africa. The death toll, mainly in Indonesia and Sri Lanka, exceeded 200,000. Nine months later, hurricane Katrina devastated the southern coast of USA along the Gulf coast. Winds reached 281 kilometers per hour and the storm surge of over nine meters was the highest recorded in the United States. It brought destruction to New Orleans when portions of the 563 kilometers of levees surrounding the city were suddenly breached. Nearly 1700 people died and damages are currently estimated at $100 billion, the costliest natural disaster in the United States. Within days hurricane Rita, another maximum category hurricane, struck the same coastal region damaging Texas and other states, followed soon aft- wards by hurricane Wilma. Then on October 8th 2005 an earthquake in Kashmir, part of northern Pakistan and India, killed 75,000 inhabitants when innumerable buildings collapsed. Simultaneously, hurricane Stan led to costly landslides and more than 2000 deaths in Central America. To highlight the major catastrophes of nature during the previous decade, Cyclone Gorky and its storm surge caused 139,000 deaths in coastal Bangladesh during 1991.


Table of Contents

Prefacep. ix
1 Univariate Extreme Value theoryp. 1
1.1 Order Statisticsp. 1
1.1.1 Distribution of the smallest valuep. 2
1.1.2 Distribution of the largest valuep. 4
1.1.3 General distributions of order statisticsp. 5
1.1.4 Plotting positionsp. 9
1.2 Extreme Value theoryp. 11
1.2.1 "Block" modelp. 12
1.2.2 "Threshold" modelp. 31
1.2.3 Scaling of extremesp. 39
1.2.4 Contagious Extreme Value distributionsp. 50
1.3 Hazard, return period, and riskp. 53
1.4 Natural Hazardsp. 58
1.4.1 Earthquakesp. 58
1.4.2 Volcanic eruptionsp. 66
1.4.3 Tsunamisp. 68
1.4.4 Landslidesp. 72
1.4.5 Avalanchesp. 77
1.4.6 Windstormsp. 83
1.4.7 Extreme sea levels and high wavesp. 86
1.4.8 Low flows and droughtsp. 89
1.4.9 Floodsp. 94
1.4.10 Wildfiresp. 106
2 Multivariate Extreme Value theoryp. 113
2.1 Multivariate Extreme Value distributionsp. 114
2.2 Characterization of the domain of attractionp. 120
2.3 Multivariate dependencep. 123
2.4 Multivariate return periodsp. 126
3 Bivariate Analysis via Copulasp. 131
3.1 2-Copulasp. 131
3.2 Archimedean copulasp. 142
3.3 Return periods via copulasp. 148
3.3.1 Univariate vsp. 150
3.3.2 The "Or" casep. 155
3.3.3 The "And" casep. 157
3.3.4 Conditional return periodsp. 159
3.3.5 Secondary return periodp. 161
3.4 Tail dependencep. 170
4 Multivariate Analysis via copulasp. 177
4.1 Multivariate copulasp. 178
4.2 Archimedean copulasp. 183
4.3 Conditional mixturesp. 186
4.3.1 The 3-Dimensional casep. 186
4.3.2 The 4-Dimensional Casep. 189
4.3.3 The general casep. 190
5 Extreme Value Analysis via Copulasp. 191
5.1 Extreme Value copulasp. 191
5.2 Dependence functionp. 201
5.3 Tail dependencep. 207
Appendix A Simulation of Copulasp. 209
A.1 The 2-Dimensional casep. 209
A.2 The general casep. 212
Appendix B Dependencep. 219
B.1 Bivariate concepts of dependencep. 219
B.1.1 Quadrant dependencep. 219
B.1.2 Tail monotonicityp. 221
B.1.3 Stochastic monotonicityp. 223
B.1.4 Corner set monotonicityp. 224
B.1.5 Dependence orderingsp. 225
B.1.6 Measure of dependencep. 225
B.2 Measures of associationp. 227
B.2.1 Measures of concordancep. 227
B.2.2 Kendall's ¿kp. 228
B.2.3 Spearman's ¿sp. 230
Appendix C Families of Copulasp. 233
C.1 The Frank familyp. 233
C.2 The Gumbel-Hougaard familyp. 236
C.3 The Clayton familyp. 237
C.4 The Ali-Mikhail-Haq (AMH) familyp. 240
C.5 The Joe familyp. 242
C.6 The Farlie-Gumbel-Morgenstern (FGM) familyp. 244
C.7 The Plackett familyp. 247
C.8 The Raftery familyp. 248
C.9 The Galambos familyp. 250
C.10 The Hüsler-Reiss familyp. 252
C.11 The Elliptical familyp. 254
C.12 The Fréchet familyp. 256
C.13 The Marshall-Olkin familyp. 257
C.14 The Archimax familyp. 264
C.15 Construction methods for copulasp. 264
C.15.1 Transformation of copulasp. 265
C.15.2 Composition of copulasp. 266
C.15.3 Copulas with given diagonal sectionp. 268
Referencesp. 271
Indexp. 285
Glossaryp. 291
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