Extreme Value and Related Models with Applications in Engineering and Science

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

• ISBN13: 9780471671725
• ISBN10: 047167172X
• Edition: 1st
• Format: Hardcover
• Copyright: 2004-11-04
• Publisher: Wiley-Interscience

Summary

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.

Author Biography

ENRIQUE CASTILLO, PhD, is a Professor of Applied Mathematics at the University of Cantabria in Santander, Spain. He is a mathematician and a civil engineer and member of the Spanish Royal Academy of Engineering. <BR> ALI S. HADI, PhD, is a Professor of Mathematical, Statistical, and Computational Sciences at the American University in Cairo, Egypt. He is a Stephen H. Weiss Presidential Fellow and Professor Emeritus at Cornell University. <BR> N. BALAKRISHNAN, PhD, is a Professor in the Department of Mathematics and Statistics at McMaster University in Ontario, Canada. He is a Fellow of the American Statistical Association and currently the Editor-in-Chief of Communications in Statistics and Wiley&#146;s Encyclopedia of Statistical Sciences, Second Edition. <BR> JOS&Eacute; M. SARABIA, PhD, is a Professor of Statistics in the Department of Economics at the University of Cantabria, Spain. He is a member of the American Statistical Association and is Associate Editor of the journal Test.

Table of Contents

Preface	xiii
I Data, Introduction, and Motivation	1	(18)
1 Introduction and Motivation	3	(16)
1.1 What Are Extreme Values?	3	(1)
1.2 Why Are Extreme Value Models Important?	4	(1)
1.3 Examples of Applications	5	(4)
1.3.1 Ocean Engineering	5	(1)
1.3.2 Structural Engineering	5	(1)
1.3.3 Hydraulics Engineering	6	(1)
1.3.4 Meteorology	7	(1)
1.3.5 Material Strength	7	(1)
1.3.6 Fatigue Strength	7	(1)
1.3.7 Electrical Strength of Materials	8	(1)
1.3.8 Highway Traffic	8	(1)
1.3.9 Corrosion Resistance	8	(1)
1.3.10 Pollution Studies	9	(1)
1.4 Univariate Data Sets	9	(6)
1.4.1 Wind Data	9	(1)
1.4.2 Flood Data	9	(1)
1.4.3 Wave Data	10	(1)
1.4.4 Oldest Age at Death in Sweden Data	10	(1)
1.4.5 Houmb's Data	10	(1)
1.4.6 Telephone Calls Data	11	(1)
1.4.7 Epicenter Data	12	(1)
1.4.8 Chain Strength Data	12	(1)
1.4.9 Electrical Insulation Data	12	(1)
1.4.10 Fatigue Data	13	(1)
1.4.11 Precipitation Data	13	(1)
1.4.12 Bilbao Wave Heights Data	13	(2)
1.5 Multivariate Data Sets l5i
1.5.1 Ocmulgee River Data	15	(1)
1.5.2 The Yearly Maximum Wind Data	15	(1)
1.5.3 The Maximum Car Speed Data	15	(4)
II Probabilistic Models Useful for Extremes	19	(86)
2 Discrete Probabilistic Models	21	(22)
2.1 Univariatc Discrete Random Variables	22	(4)
2.1.1 Probability Mass Function	22	(1)
2.1.2 Cumulative Distribution Function	23	(1)
2.1.3 Moments	24	(2)
2.2 Common Univariate Discrete Models	26	(14)
2.2.1 Discrete Uniform Distribution	26	(1)
2.2.2 Bernoulli Distribution	26	(2)
2.2.3 Binomial Distribution	28	(3)
2.2.4 Geometric or Pascal Distribution	31	(2)
2.2.5 Negative Binomial Distribution	33	(2)
2.2.6 Hypergcometric Distribution	35	(1)
2.2.7 Poisson Distribution	36	(3)
2.2.8 Nonzero Poisson Distribution	39	(1)
Exercises	40	(3)
3 Continuous Probabilistic Models	43	(42)
3.1 Univariate Continuous Random Variables	43	(3)
3.1.1 Probability Density Function	43	(1)
3.1.2 Cumulative Distribution Function	44	(1)
3.1.3 Moments	45	(1)
3.2 Common Univariate Continuous Models	46	(20)
3.2.1 Continuous Uniform Distribution	46	(1)
3.2.2 Exponential Distribution	47	(2)
3.2.3 Gamma Distribution	49	(4)
3.2.4 Log-Gamma Distribution	53	(1)
3.2.5 Beta Distribution	54	(1)
3.2.6 Normal or Gaussian Distribution	55	(4)
3.2.7 Log-Normal Distribution	59	(1)
3.2.8 Logistic Distribution	59	(1)
3.2.9 Chi-Square and Chi Distributions	60	(1)
3.2.10 Rayleigh Distribution	61	(1)
3.2.11 Student's t Distribution	61	(1)
3.2.12 F Distribution	61	(1)
3.2.13 Weibull Distribution	62	(1)
3.2.14 Gumbel Distribution	63	(1)
3.2.15 Fréchet Distribution	63	(1)
3.2.16 Generalized Extreme Value Distributions	64	(1)
3.2.17 Generalized Pareto Distributions	65	(1)
3.3 Truncated Distributions	66	(6)
3.4 Some Other Important Functions	72	(9)
3.4.1 Survival and Hazard Functions	72	(2)
3.4.2 Moment Generating Function	74	(2)
3.4.3 Characteristic Function	76	(5)
Exercises	81	(4)
4 Multivariate Probabilistic Models	85	(20)
4.1 Multivariate Discrete Random Variables	85	(5)
4.1.1 Joint Probability Mass Function	85	(1)
4.1.2 Marginal Probability Mass Function	86	(1)
4.1.3 Conditional Probability Mass Function	86	(1)
4.1.4 Covariance and Correlation	87	(3)
4.2 Common Multivariate Discrete Models	90	(2)
4.2.1 Multinomial Distribution	91	(1)
4.2.2 Multivariate Hypergeometric Distribution	92	(1)
4.3 Multivariate Continuous Random Variables	92	(6)
4.3.1 Joint Probability Density Function	93	(1)
4.3.2 Joint Cumulative Distribution Function	93	(1)
4.3.3 Marginal Probability Density Functions	94	(1)
4.3.4 Conditional Probability Density Functions	94	(1)
4.3.5 Covariance and Correlation	95	(1)
4.3.6 The Autocorrelation Function	96	(1)
4.3.7 Bivariate Survival and Hazard Functions	96	(2)
4.3.8 Bivariate CDF and Survival Function	98	(1)
4.3.9 Joint Characteristic Function	98	(1)
4.4 Common Multivariate Continuous Models	98	(3)
4.4.1 Bivariate Logistic Distribution	98	(1)
4.4.2 Multinormal Distribution	99	(1)
4.4.3 Marshall-Olkin Distribution	99	(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