1 Introduction
1.1 Basic Concepts
1.2 Examples
1.3 Structme of the Book
1.4 Further Reading
2 Basics of Statistical Modeling
2.1 Introduction
2.2 Basic Statistical Concepts
2.2.1 Random Variables and Their Distributions
2.2.2 Families of Models
2.3 Multivariate Distributions
2.4 Random Processes
2.4.1 Stationary Pro(es,,es
2.4.2 Markov Chains
2.5 Limit Laws
2.6 Parametric Modeling
2.6.1 The Parametric Framework
2.6.2 Principles ef Estimaticn
2.6.3 Maximum Likelihood Estimation
2.6.4 Approximate Normality cf the Maximum Likelihood Estimator
2.6.5 Approximate Inference Using the Deviance Function
2.6.6 Inference Using the Profile Likelihood Function
2.6.7 Model Diagnostics
2.7 Example
2.8 Further Reading
3 Classical Extreme Value Theory and Models
3.1 Asymptotic Models
3.1.1 Model Formulation
3.1.2 Extremal Types Theorem
3.1.3 The Generalized Extreme Value Distribution
3.1.4 Outline Proof of the Extremal Types Theorem
3.1.5 Examples
3.2 Asymptotic Models for Minima
3.3 Inference for the GEV Distribution
3.3.1 General Considerations
3.3.2 Maximum Likelihood Estimation
3.3.3 Inference for Return Levels
3.3.4 Profile Likelihood
3.3.5 Model Checking
3.4 Examples
3.4.1 Annual Maximum Sea-levels at Port Pirie
3.4.2 Glass Fiber Strength Example
3.5 Model Generalization: the r Largest Order Statistic Model
3.5.1 Model Formulation
3.5.2 Modeling the r Largest Order Statistics
3.5.3 Venice Sea-level Data
3.6 Further Reading
4 Threshold Models
4.1 Introduction
4.2 Asymptotic Model Characterization
4.2.1 The Generalized Pareto Distribution
4.2.2 Outline Justification for the Generalized Pareto Model
4.2.3 Examples
4.3 Modeling Threshold Excesses
4.3.1 Threshold Selection
4.3.2 Parameter Estimation
4.3.3 Return Levels
4.3.4 Threshold Choice Revisited
4.3.5 Model Checking
4.4 Examples
4.4.1 Daily Rainfall Data
4.4.2 Dow Jones Index Series
4.5 Further Reading
5 Extremes of Dependent Sequences
5.1 Introduction
5.2 Maxima of Stationary Sequences
5.3 Modeling Stationary Series
5.3.1 Models for Block Maxima
5.3.2 Threshold Models
5.3.3 Wooster Temperature Series
5.3.4 Dow Jones Index Series
5.4 Further Reading
6 Extremes of Non-stationary Sequences
6.1 Model Structures
6.2 Inference
6.2.1 Parameter Estimation
6.2.2 Model Choice
6.2.3 Model Diagnostics
6.3 Examples
6.3.1 Annual Maximum Sea-levels
6.3.2 Race Time Data
6.3.3 Venice Sea-level Data
6.3.4 Daily Rainfall Data
6.3.5 Wooster Temperature Data
6.4 Further Reading
7 A Point Process Characterization of Extremes
7.1 Introduction
7.2 Basic Theory of Point Processes
7.3 A Poisson Process Limit for Extremes
7.3.1 Convergence Law
7.3.2 Examples
7.4 Connections with Other Extreme Value Models
7.5 Statistical Modeling
7.6 Connections with Threshold Excess Model Likelihood
7.7 Wooster Temperature Series
7.8 Return Level Estimation
7.9 Largest Order Statistic Model
7.10 Further Reading
8 Multivariate Extremes
8.1 Introduction
8.2 Componentwise Maxima
8.2.1 Asymptotic Characterization
8.2.2 Modeling
8.2.3 Example: Annual Maximum Sea-levels
8.2.4 Structure Variables
8.3 Alternative Representations
8.3.1 Bivariate Threshold Excess Model
8.3.2 Point Process Model
8.3.3 Examples
8.4 Asymptotic Independence
8.5 Further Reading
9 Further Topics
9.1 Bayesian Inference
9.1.1 General Theory
9.1.2 Bayesian Inference of Extremes
9.1.3 Example: Port Pirie Annual Maximum Sea-levels
9.2 Extremes of Markov Chains
9.3 Spatial Extremes
9.3.1 Max-stable Processes
9.3.2 Latent Spatial Process Models
9.4 Further Reading
A Computational Aspects
References
Index