JOINT-PARAMETER ESTIMATION, BOOTSTRAP BIAS CORRECTION AND RISK FORECAST FOR EXTREME EVENTS

Abstract

Extreme value theory is a branch of mathematics that studies extreme events. Given a series modeling the event of interest, two quantities are of particular importance. One is the shape parameter, which detects the presence of heavy tails in the data; the other is the extremal index, which measures the degree of clustering of the extremes. This thesis is dedicated to the estimation and proper application of the two quantities to better understand the extreme events. The topics are developed through 3 parts. In the first part, we start by establishing the functional joint asymptotic normality of the common estimators of the shape parameter and the extremal index. This improves the previous works in which the two quantities are estimated independently from each other. We then propose an approach to forecast risk contained in future observations in a time series by incorporating both the shape parameter and the extremal index. This significantly improves the quality of risk forecasting over methods that are designed for i.i.d. observations and over the return level approach. In the second part, we propose a regression and bootstrap based technique to reduce the bias of the Hill estimator. While most of the existing bias correction techniques are based on i.i.d. data, our work covers the missing yet important case of dependent data. In simulation studies, our bias corrected Hill estimator demonstrates significant improvements in bias and mean squared error (MSE) over the uncorrected Hill estimator. In the last part, we extend the joint estimation of the parameters and the risk forecast procedure from weakly dependent time series to strongly mixing random fields.