Extreme Value Theory An Introduction

Transcription

1 Laurens de Haan Ana Ferreira Extreme Value Theory An Introduction fi Springer

2 Contents Preface List of Abbreviations and Symbols vii xv Part I One-Dimensional Observations 1 Limit Distributions and Domains of Attraction Extreme Value Theory: Basics Introduction Alternative Formulations of the Limit Relation Extreme Value Distributions Interpretation of the Alternative Conditions; Case Studies Domains of Attraction: A First Approach Domains of Attraction 19 Exercises 34 2 Extreme and Intermediate Order Statistics Extreme Order Statistics and Poisson Point Processes Intermediate Order Statistics Second-Order Condition Intermediate Order Statistics and Brownian Motion 49 Exercises 60 3 Estimation of the Extreme Value Index and Testing Introduction A Simple Estimator for the Tail Index (y > 0): The Hill Estimator General Case y e R: The Pickands Estimator The Maximum Likelihood Estimator (y > \) A Moment Estimator (y e R) Other Estimators The Probability-Weighted Moment Estimator (y < 1) 110

3 xii Contents The Negative Hill Estimator (y < -\) Simulations and Applications Asymptotic Properties Simulations Case Studies 121 Exercises Extreme Quantile and Tail Estimation Introduction Scale Estimation Quantile Estimation Maximum Likelihood Estimators Moment Estimators Tail Probability Estimation Maximum Likelihood Estimators Moment Estimators Endpoint Estimation Maximum Likelihood Estimators Moment Estimators Simulations and Applications Simulations Case Studies 149 Exercises Advanced Topics Expansion of the Tail Distribution Function and Tail Empirical Process Checking the Extreme Value Condition Convergence of Moments, Speed of Convergence, and Large Deviations Convergence of Moments Speed of Convergence; Large Deviations Weak and Strong Laws of Large Numbers and Law of the Iterated Logarithm Weak "Temporal" Dependence Mejzler's Theorem 201 Exercises 204 Part II Finite-Dimensional Observations 6 Basic Theory Limit Laws Introduction: An Example The Limit Distribution; Standardization The Exponent Measure 211

4 Contents xiii The Spectral Measure The Sets Q c and the Functions L, x, and A Domains of Attraction; Asymptotic Independence 226 Exercises Estimation of the Dependence Structure Introduction Estimation of the Function L and the Sets Q c Estimation of the Spectral Measure (and L) A Dependence Coefficient Tail Probability Estimation and Asymptotic Independence: A Simple Case Estimation of the Residual Dependence Index r\ 265 Exercises Estimation of the Probability of a Failure Set Introduction Failure Set with Positive Exponent Measure First Approach: c n Known Alternative Approach: Estimate c n Proofs Failure Set Contained in an Upper Quadrant; Asymptotically Independent Components Sea Level Case Study 288 Exercises 289 Part in Observations That Are Stochastic Processes 9 Basic Theory in C[0,1] Introduction: An Example The Limit Distribution; Standardization The Exponent Measure The Spectral Measure Domain of Attraction Spectral Representation and Stationarity Spectral Representation Stationarity Special Cases Two Examples 323 Exercises Estimation in C[0,1] Introduction: An Example Estimation of the Exponent Measure: A Simple Case Estimation of the Exponent Measure 335

5 xiv Contents 10.4 Estimation of the Index Function, Scale and Location Consistency Asymptotic Normality Estimation of the Probability of a Failure Set 349 Part IV Appendix A Skorohod Theorem and Vervaat's Lemma 357 B Regulär Variation and Extensions 361 B.l Regularly Varying (RV) Functions 361 B.2 Extended Regulär Variation (ERV); The class n 371 B.3 Second-Order Extended Regulär Variation (2ERV) 385 B.4 ERV with an Extra Parameter 401 References 409 Index 415!