Extreme Value and Related Models with Applications in Engineering and Science

, by Castillo, Enrique; Hadi, Ali S.; Balakrishnan, Narayanaswamy; Sarabia, Jose M.

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ISBN: 9780471671725 | 047167172X
Cover: Hardcover
Copyright: 11/4/2004

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Here's an elementary and comprehensive discussion on extreme value and related models. By using a large number of practical data from different science and engineering disciplines, it illustrates the practical importance and usefulness of extreme value modeling. Unusual, provocative, and concrete examples are derived from areas such as ocean engineering, structural engineering, hydraulics, meteorology, materials science, fatigue studies, electrical strength of materials, highway traffic analysis, corrosion science, environmetrics, climatology, among others. * Emphasizes lucid and analytic explanations of the data in the context of a global world. * Presents specific data in wind, flood, wave, Houmb's, Ocmulgee River, oldest age at death in Sweden, telephone calls, epicenter, chain strength, electrical insulation, fatigue, precipitation, and Bilbao wave heights phenomena. * Discusses different types of inference, extreme value regression, and handling of outliers.

Preface

xiii

I Data, Introduction, and Motivation

(18)

1 Introduction and Motivation

(16)

1.1 What Are Extreme Values?

(1)

1.2 Why Are Extreme Value Models Important?

(1)

1.3 Examples of Applications

(4)

1.3.1 Ocean Engineering

(1)

1.3.2 Structural Engineering

(1)

1.3.3 Hydraulics Engineering

(1)

1.3.4 Meteorology

(1)

1.3.5 Material Strength

(1)

1.3.6 Fatigue Strength

(1)

1.3.7 Electrical Strength of Materials

(1)

1.3.8 Highway Traffic

(1)

1.3.9 Corrosion Resistance

(1)

1.3.10 Pollution Studies

(1)

1.4 Univariate Data Sets

(6)

1.4.1 Wind Data

(1)

1.4.2 Flood Data

(1)

1.4.3 Wave Data

(1)

1.4.4 Oldest Age at Death in Sweden Data

(1)

1.4.5 Houmb's Data

(1)

1.4.6 Telephone Calls Data

(1)

1.4.7 Epicenter Data

(1)

1.4.8 Chain Strength Data

(1)

1.4.9 Electrical Insulation Data

(1)

1.4.10 Fatigue Data

(1)

1.4.11 Precipitation Data

(1)

1.4.12 Bilbao Wave Heights Data

(2)

1.5 Multivariate Data Sets l5i

1.5.1 Ocmulgee River Data

(1)

1.5.2 The Yearly Maximum Wind Data

(1)

1.5.3 The Maximum Car Speed Data

(4)

II Probabilistic Models Useful for Extremes

(86)

2 Discrete Probabilistic Models

(22)

2.1 Univariatc Discrete Random Variables

(4)

2.1.1 Probability Mass Function

(1)

2.1.2 Cumulative Distribution Function

(1)

2.1.3 Moments

(2)

2.2 Common Univariate Discrete Models

(14)

2.2.1 Discrete Uniform Distribution

(1)

2.2.2 Bernoulli Distribution

(2)

2.2.3 Binomial Distribution

(3)

2.2.4 Geometric or Pascal Distribution

(2)

2.2.5 Negative Binomial Distribution

(2)

2.2.6 Hypergcometric Distribution

(1)

2.2.7 Poisson Distribution

(3)

2.2.8 Nonzero Poisson Distribution

(1)

Exercises

(3)

3 Continuous Probabilistic Models

(42)

3.1 Univariate Continuous Random Variables

(3)

3.1.1 Probability Density Function

(1)

3.1.2 Cumulative Distribution Function

(1)

3.1.3 Moments

(1)

3.2 Common Univariate Continuous Models

(20)

3.2.1 Continuous Uniform Distribution

(1)

3.2.2 Exponential Distribution

(2)

3.2.3 Gamma Distribution

(4)

3.2.4 Log-Gamma Distribution

(1)

3.2.5 Beta Distribution

(1)

3.2.6 Normal or Gaussian Distribution

(4)

3.2.7 Log-Normal Distribution

(1)

3.2.8 Logistic Distribution

(1)

3.2.9 Chi-Square and Chi Distributions

(1)

3.2.10 Rayleigh Distribution

(1)

3.2.11 Student's t Distribution

(1)

3.2.12 F Distribution

(1)

3.2.13 Weibull Distribution

(1)

3.2.14 Gumbel Distribution

(1)

3.2.15 Fréchet Distribution

(1)

3.2.16 Generalized Extreme Value Distributions

(1)

3.2.17 Generalized Pareto Distributions

(1)

3.3 Truncated Distributions

(6)

3.4 Some Other Important Functions

(9)

3.4.1 Survival and Hazard Functions

(2)

3.4.2 Moment Generating Function

(2)

3.4.3 Characteristic Function

(5)

Exercises

(4)

4 Multivariate Probabilistic Models

(20)

4.1 Multivariate Discrete Random Variables

(5)

4.1.1 Joint Probability Mass Function

(1)

4.1.2 Marginal Probability Mass Function

(1)

4.1.3 Conditional Probability Mass Function

(1)

4.1.4 Covariance and Correlation

(3)

4.2 Common Multivariate Discrete Models

(2)

4.2.1 Multinomial Distribution

(1)

4.2.2 Multivariate Hypergeometric Distribution

(1)

4.3 Multivariate Continuous Random Variables

(6)

4.3.1 Joint Probability Density Function

(1)

4.3.2 Joint Cumulative Distribution Function

(1)

4.3.3 Marginal Probability Density Functions

(1)

4.3.4 Conditional Probability Density Functions

(1)

4.3.5 Covariance and Correlation

(1)

4.3.6 The Autocorrelation Function

(1)

4.3.7 Bivariate Survival and Hazard Functions

(2)

4.3.8 Bivariate CDF and Survival Function

(1)

4.3.9 Joint Characteristic Function

(1)

4.4 Common Multivariate Continuous Models

(3)

4.4.1 Bivariate Logistic Distribution

(1)

4.4.2 Multinormal Distribution

(1)

4.4.3 Marshall-Olkin Distribution

(1)

4.4.4 Freund's Bivariate Exponential Distribution

100

(1)

Exercises

101

(4)

III Model Estimation, Selection, and Validation

105

(46)

5 Model Estimation

107

(26)

5.1 The Maximum Likelihood Method

108

(9)

5.l.l Point Estimation

108

(2)

5.1.2 Some Properties of the MLE

110

(2)

5.1.3 The Delta Method

112

(1)

5.1.4 Interval Estimation

113

(1)

5.1.5 The Deviance Function

114

(3)

5.2 The Method of Moments

117

(1)

5.3 The Probability-Weighted Moments Method

117

(2)

5.4 The Elemental Percentile Method

119

(3)

5.4.1 Initial Estimates

120

(1)

5.4.2 Confidence Intervals

121

(1)

5.5 The Quantile Least Squares Method

122

(1)

5.6 The Truncation Method

123

(1)

5.7 Estimation for Multivariate Models

123

(6)

5.7.1 The Maximum Likelihood Method

123

(2)

5.7.2 The Weighted Least Squares CDF Method

125

(1)

5.7.3 The Elemental Percentile Method

125

(1)

5.7.4 A Method Based on Least Squares

126

(3)

Exercises

129

(4)

6 Model Selection and Validation

133

(18)

6.1 Probability Paper Plots

134

(12)

6.1.1 Normal Probability Paper Plot

137

(1)

6.1.2 Log-Normal Probability Paper Plot

138

(3)

6.1.3 Gumbel Probability Paper Plot

141

(1)

6.1.4 Weibull Probability Paper Plot

142

(4)

6.2 Selecting Models by Hypothesis Testing

146

(2)

6.3 Model Validation

148

(1)

6.3.1 The Q-Q Plots

148

(1)

6.3.2 The P-P Plots

148

(1)

Exercises

149

(2)

IV Exact Models for Order Statistics and Extremes

151

(40)

7 Order Statistics

153

(24)

7.1 Order Statistics and Extremes

153

(1)

7.2 Order Statistics of Independent Observations

153

(11)

7.2.1 Distributions of Extremes

154

(3)

7.2.2 Distribution of a Subset of Order Statistics

157

(1)

7.2.3 Distribution of a Single Order Statistic

158

(4)

7.2.4 Distributions of Other Special Cases

162

(2)

7.3 Order Statistics in a Sample of Random Size

164

(2)

7.4 Design Values Based on Exceedances

166

(2)

7.5 Return Periods

168

(2)

7.6 Order Statistics of Dependent Observations

170

(3)

7.6.1 The Inclusion-Exclusion Formula

170

(1)

7.6.2 Distribution of a Single Order Statistic

171

(2)

Exercises

173

(4)

8 Point Processes and Exact Models

177

(14)

8.1 Point Processes

177

(4)

8.2 The Poisson-Flaws Model

181

(2)

8.3 Mixture Models

183

(1)

8.4 Competing Risk Models

184

(1)

8.5 Competing Risk Flaws Models

185

(1)

8.6 Poissonian Storm Model

186

(2)

Exercises

188

(3)

V Asymptotic Models for Extremes

191

(134)

9 Limit Distributions of Order Statistics

193

(68)

9.1 The Case of Independent Observations

193

(18)

9.1.1 Limit Distributions of Maxima and Minima

194

(4)

9.1.2 Weibull, Gumbel, and Frechet as GEVDs

198

(2)

9.1.3 Stability of Limit Distributions

200

(3)

9.1.4 Determining the Domain of Attraction of a CDF

203

(5)

9.1.5 Asymptotic Distributions of Order Statistics

208

(3)

9.2 Estimation for the Maximal GEVD

211

(15)

9.2.1 The Maximum Likelihood Method

212

(6)

9.2.2 The Probability Weighted Moments Method

218

(2)

9.2.3 The Elemental Percentile Method

220

(4)

9.2.4 The Quantile Least Squares Method

224

(1)

9.2.5 The Truncation Method

225

(1)

9.3 Estimation for the Minimal GEVD

226

(1)

9.4 Graphical Methods for Model Selection

226

(10)

9.4.1 Probability Paper Plots for Extremes

228

(6)

9.4.2 Selecting a Domain of Attraction from Data

234

(2)

9.5 Model Validation

236

(1)

9.6 Hypothesis Tests for Domains of Attraction

236

(12)

9.6.1 Methods Based on Likelihood

243

(2)

9.6.2 The Curvature Method

245

(3)

9.7 The Case of Dependent Observations

248

(10)

9.7.1 Stationary Sequences

249

(3)

9.7.2 Exchangeable Variables

252

(2)

9.7.3 Markov Sequences of Order p

254

(1)

9.7.4 The m-Dependent Sequences

254

(1)

9.7.5 Moving Average Models

255

(1)

9.7.6 Normal Sequences

256

(2)

Exercises

258

(3)

10 Limit Distributions of Exceedances and Shortfalls

261

(1)

10.1 Exceedances as a Poisson Process

262

(1)

10.2 Shortfalls as a Poisson Process

262

(1)

10.3 The Maximal GPD

263

(2)

10.4 Approximations Based on the Maximal GPD

265

(1)

10.5 The Minimal GPD

266

(1)

10.6 Approximations Based on the Minimal GPD

267

(1)

10.7 Obtaining the Minimal from the Maximal GPD

267

(1)

10.8 Estimation for the GPD Families

268

(19)

10.8.1 The Maximum Likelihood Method

268

(3)

10.8.2 The Method of Moments

271

(1)

10.8.3 The Probability Weighted Moments Method

271

(1)

10.8.4 The Elemental Percentile Method

272

(4)

10.8.5 The Quantile Least Squares Method

276

(1)

10.9 Model Validation

277

(4)

10.10 Hypothesis Tests for the Domain of Attraction

281

(4)

Exercises

285

(2)

11 Multivariate Extremes

287

(38)

11.1 Statement of the Problem

288

(1)

11.2 Dependence Functions

289

(2)

11.3 Limit Distribution of a Given CDF

291

(7)

11.3.1 Limit Distributions Based on Marginals

291

(4)

11.3.2 Limit Distributions Based on Dependence Functions

295

(3)

11.4 Characterization of Extreme Distributions

298

(6)

11.4.1 Identifying Extreme Value Distributions

299

(1)

11.4.2 Functional Equations Approach

299

(1)

11.4.3 A Point Process Approach

300

(4)

11.5 Some Parametric Bivariate Models

304

(1)

11.6 Transformation to Frechet Marginals

305

(1)

11.7 Peaks Over Threshold Multivariate Model

306

(1)

11.8 Inference

307

(2)

11.8.1 The Sequential Method

307

(1)

11.8.2 The Single Step Method

308

(1)

11.8.3 The Generalized Method

309

(1)

11.9 Sonic Multivariate Examples

309

(9)

11.9.1 The Yearly Maximum Wind Data

309

(3)

11.9.2 The Ocmulgee River Flood Data

312

(4)

11.9.3 The Maximum Car Speed Data

316

(2)

Exercises

318

(7)

Appendix A: Statistical Tables

325

(8)

Bibliography

333

(20)

Index

353

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Extreme Value and Related Models with Applications in Engineering and Science

Summary

Author Biography

Table of Contents

Supplemental Materials