Extreme value theory (EVT) deals with extreme (rare) events, which are sometimes reported as outliers. Certain textbooks encourage readers to remove outliers--in other words, to correct reality if it does not fit the model. Recognizing that any model is only an approximation of reality, statisticians are eager to extract information about unknown distribution making as few assumptions as possible. Extreme Value Methods with Applications to Finance concentrates on modern topics in EVT, such as processes of exceedances, compound Poisson approximation, Poisson cluster approximation, and nonparametric estimation methods. These topics have not been fully focused on in other books on extremes. In addition, the book covers: Extremes in samples of random size Methods of estimating extreme quantiles and tail probabilities Self-normalized sums of random variables Measures of market risk Along with examples from finance and insurance to illustrate the methods, Extreme Value Methods with Applications to Finance includes over 200 exercises, making it useful as a reference book, self-study tool, or comprehensive course text. A systematic background to a rapidly growing branch of modern Probability and Statistics: extreme value theory for stationary sequences of random variables.
Introduction Distribution of Extremes Methods of Extreme Value Theory Order Statistics "Blocks" and "Runs" Approaches Method of Recurrent Inequalities Proofs Maximum of Partial Sums Erdos--Renyi Maximum of Partial Sums Basic Inequalities Limit Theorems for MPS Proofs Extremes in Samples of Random Size Maximum of a Random Number of r.v.s Number of Exceedances Length of the Longest Head Run Long Match Patterns Poisson Approximation Total Variation Distance Method of a Common Probability Space The Stein Method Beyond Bernoulli The Magic Factor Proofs Compound Poisson Approximation Limit Theory Accuracy of CP Approximation Proofs Exceedances of Several Levels CP Limit Theory General Case Accuracy of Approximation Proofs Processes of Exceedances One-level EPPE Excess Process Complete Convergence to CP Processes Proofs Beyond Compound Poisson Excess Process Complete Convergence Proofs Statistics of Extremes Inference on Heavy Tails Heavy-tailed distributions Estimation Methods Tail Index Estimation Estimation of Extreme Quantiles Estimation of the Tail Probability Proofs Value-at-Risk. Value-at-Risk and Expected Shortfall Traditional Methods of VaR Estimation VaR and ES Estimation from Heavy-Tailed Data VaR over Different Time Horizons Technical Analysis of Financial Data Extremal Index Preliminaries Estimation of the Extremal Index Proofs Normal Approximation. Accuracy of Normal Approximation Stein's Method Self-Normalized Sums of r.v.s Proofs Lower Bounds Preliminary Results Frechet--Rao--Cramer Inequality Information Index Continuity Moduli Tail Index and Extreme Quantiles Proofs Appendix Probability Distributions Properties of Distributions Probabilistic Identities and Inequalities Distances Large Deviations Elements of Renewal Theory Dependence Point Processes Slowly Varying Functions Useful Identities and Inequalities References Index
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