EVALUATION OF SEMI-PARAMETRIC ESTIMATION METHODS OF THE EXTREME-VALUE INDEX AND ROBUSTIFYING MODIFICATIONS


 

 

        Tsourti Zoi
        Supervisor: I. Panaretos
 

        CHAPTER 1
        INTRODUCTION 

        1.1 Preface
        1.2 Genesis and Historical Development
        1.3 Fields of Application

        1.4 Overview

  

        CHAPTER 2

        THEORETICAL BACKGROUND

        2.1 Introduction
        2.2 Theory of Regular Variation
        2.3 Limit Laws for Maxima

        2.4 Generalized Extreme-Value Distribution

        2.5 Limit Laws for Minima

        2.6 Generalized Pareto Distribution

        2.7 Examples and Counter-Examples

 

        CHAPTER 3
        PARAMETRIC APPROACH TO MODELLING EXTREMES

        3.1 Introduction

        3.2 Maximum Likelihood Estimation

        3.3 Method of Probability Weighted Moments

        3.4 Comparison of ML and PWM Estimation Methods for GEV Distribution

        3.5 Modified ML Estimators

        3.6 Other Estimation Techniques

 

        CHAPTER 4
        SEMI-PARAMETRIC APPROACH TO MODELLING EXTREMES

        4.1 Introduction

        4.2 Pickands Estimator

            4.2.1 Derivation

            4.2.2 Choice of k

            4.2.3 Properties of Pickands Estimator

            4.2.4 Quantile estimation

            4.2.5 Modifications - Developments on Pickands Estimator

        4.3 Hill Estimator

            4.3.1 Derivation

            4.3.2 Choice of k

            4.3.3 Properties of Hill Estimator

            4.3.4 Quantile estimation

            4.3.5 Modifications - Developments on Hill Estimator

            4.3.6 Asymptotic Behaviour of Hill Estimator Based on Dependent Data

        4.4 Adapted Hill Estimator

        4.5 Moment Estimator

            4.5.1 Derivation

            4.5.2 Properties of Moment Estimator

            4.5.3 Quantile estimation

        4.6 Other Semi-Parametric Estimation Methods

            4.6.1 Moments Ratio Estimator

            4.6.2 Kernel Estimators

            4.6.3 QQ-Estimator

            4.6.4 "k-records" Estimator

            4.6.5 Other Extreme-Value Index Estimators

        4.7 "Peaks Over Thresholds" Estimation Methods

        CHAPTER 5

        SMOOTHING AND ROBUSTIFYING PROCEDURES FOR SEMI-PARAMETRIC

        EXTREME-VALUE INDEX ESTIMATORS

        5.1 Introduction

        5.2 Smoothing Extreme-Value Index Estimators

            5.2.1 Smoothing Hill Estimator

            5.2.2 Smoothing Moment Estimator

        5.3 Robust Estimators Based on Excess Plots

        5.4 Simulation Comparison of Extreme-Value Index Estimators

            5.4.1 Details of Simulation Study

            5.4.2 Discussion of Simulation Results

        5.5 Methods of Selecting k

            5.5.1 Regression Approach

            5.5.2 Bootstrap Approach

        CHAPTER 6
        EXTREME-VALUE ANALYSIS

        6.1 Introduction

        6.2 Exploratory Data Analysis

            6.2.1 Description of the Data

            6.2.2 Investigation of Independence

            6.2.3 Investigation of Heavy Tails

        6.3 Extreme-Value Analysis

            6.3.1 Estimation of Extreme-Value Index γ

            6.3.2 Estimation of Large Quantiles

        CHAPTER 7
        CONCLUSIONS

        7.1 Discussion

        7.2 Open Problems

        APPENDIX

        REFERENCES