Novak Extreme Value Methods with Applications to Finance

Erscheinungsjahr 2012
ISBN: 978-1-4398-3575-3
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 399 Seiten

Reihe: Chapman & Hall/CRC Monographs on Statistics & Applied Probability

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.

Autoren/Hrsg.

Novak, Serguei Y.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Interdisziplinär Finanz- und Versicherungsmathematik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

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–Rényi 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
Fréchét–Rao–Cramér 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

Über Autor(innen)

Dr S.Y. Novak earned his Ph.D. at the Novosibirsk Institute of Mathematics under the supervision of Dr S.A. Utev in 1988. The Novosibirsk group forms a part of Russian tradition in Probability & Statistics that extends its roots to Kolmogorov and Markov.

Dr S.Y. Novak began his teaching carrier at the Novosibirsk Electrotechnical Institute (NETI) and Novosibirsk Institute of Geodesy, held post-doctoral positions at the University of Sussex and Eurandom (Technical University of Eindhoven), and taught at Brunel University in West London, before joining the Middlesex University (London) in 2003. He published over 40 papers, mostly on the topic of Extreme Value Theory, in which he is considered an expert.