Inference for Heavy-Tailed Data ：Applications in Insurance and Finance

Publication subTitle ：Applications in Insurance and Finance

Author： Peng Liang;Qi Yongcheng

Publisher： Elsevier Science‎

Publication year： 2017

E-ISBN: 9780128047507

P-ISBN(Paperback): 9780128046760

Subject： O21 Probability and Mathematical Statistics

Keyword：数学

Language： ENG

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Description

Heavy tailed data appears frequently in social science, internet traffic, insurance and finance. Statistical inference has been studied for many years, which includes recent bias-reduction estimation for tail index and high quantiles with applications in risk management, empirical likelihood based interval estimation for tail index and high quantiles, hypothesis tests for heavy tails, the choice of sample fraction in tail index and high quantile inference. These results for independent data, dependent data, linear time series and nonlinear time series are scattered in different statistics journals. Inference for Heavy-Tailed Data Analysis puts these methods into a single place with a clear picture on learning and using these techniques.

Contains comprehensive coverage of new techniques of heavy tailed data analysis
Provides examples of heavy tailed data and its uses
Brings together, in a single place, a clear picture on learning and using these techniques

Chapter

Front Cover

Inference for Heavy-Tailed Data

Contents

About the Authors

Preface

1 Introduction

1.1 Basic Probability Theory

1.2 Basic Extreme Value Theory

2 Heavy Tailed Independent Data

2.1 Heavy Tail

2.2 Tail Index Estimation

2.2.1 Hill Estimator

2.2.1.1 Asymptotic Properties

2.2.1.2 Optimal Choice of k

2.2.1.3 Data Driven Methods for Choosing k

2.2.1.4 Bias Corrected Estimation

2.2.1.5 Sample Fraction Choice Motivated by Bias Corrected Estimation

2.2.2 Other Tail Index Estimators

2.3 High Quantile Estimation

2.4 Extreme Tail Probability Estimation

2.5 Interval Estimation

2.5.1 Conﬁdence Intervals for Tail Index

2.5.1.1 Normal Approximation Method

2.5.1.2 Bootstrap Method

2.5.1.3 Empirical Likelihood Method

2.5.2 Conﬁdence Intervals for High Quantile

2.6 Goodness-of-Fit Tests

2.7 Estimation of Mean

2.8 Expected Shortfall

2.9 Haezendonck-Goovaerts (H-G) Risk Measure

3 Heavy Tailed Dependent Data

3.1 Tail Empirical Process and Tail Quantile Process

3.2 Heavy Tailed Dependent Sequence

3.3 ARMA Model

3.4 Stochastic Difference Equations

3.5 Heavy Tailed GARCH Sequences

3.6 Double AR(1) Model

3.7 Conditional Value-at-Risk

3.8 Heavy Tailed AR-GARCH Sequences

3.9 Self-Weighted Estimation for ARMA-GARCH Models

3.10 Unit Root Tests With Inﬁnite Variance Errors

4 Multivariate Regular Variation

4.1 Multivariate Regular Variation

4.2 Hidden Multivariate Regular Variation

4.3 Tail Dependence and Extreme Risks Under Multivariate Regular Variation

4.4 Loss Given Default Under Multivariate Regular Variation

4.5 Estimating an Extreme Set Under Multivariate Regular Variation

4.6 Extreme Geometric Quantiles Under Multivariate Regular Variation

5 Applications

5.1 Some Visualization Tools for Preliminary Analysis

5.1.1 Hill Plot

5.1.2 Alternative Hill Plot

5.1.3 Log-Quantile Plot

5.2 Heuristic Approach for Training Data

5.3 Applications to Independent Data

5.3.1 Automobile Bodily Injury Claims

5.3.2 Automobile Insurance Claims

5.3.3 Hospital Costs

5.3.4 Danish Fire Losses Data

5.4 Applications to Dependent Data

5.4.1 Daily Foreign Exchange Rates

5.4.2 Quarterly S&P 500 Indices

5.4.3 S&P 500 Weighted Daily Returns

5.5 Some Comments

A Tables

B List of Notations and Abbreviations

Bibliography

Index

Back Cover

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