On Risk Measures and their Estimation

Abstract

The objective of this thesis is to apply two fundamental concepts of mathematical statistics, namely survival analysis and extreme value theory, to the estimation of risk measures. Extreme values theory provides indispensable tools for measuring the probability of unusual incidents occurring, which is a basic requirement for accurate risk estimation, even in the presence of incomplete data. We proposed an estimator of one of the most important measures of risk called the conditional tail expectation of data that are heavy-tailed and randomly censored to the right and we established its asymptotic normality. This estimation procedure is evaluated through a simulation study and applied to two real datasets of insurance losses and survival time of AIDS patients.