This dissertation consists of three essays about statistical estimation and inference methods concerning extremal events and tail risks. The first essay establishes a natural, semi-parametric estimation procedure for the multivariate half-space depth-based extreme quantile region in arbitrary dimensions. In contrast to the failure of fully non-parametric approaches due to the scarcity of extremal observations, the good finite-sample performance of our extreme estimator is clearly demonstrated in simulation studies. The second essay extends this method to various depth functions, and, furthermore, establishes an asymptotic approximation theory of what-we-called (directed) logarithmic distance between our estimated and true quantile region. Confidence sets that asymptotically cover the quantile region or its complement, or both simultaneously, with a pre-specified probability can be therefore constructed under weak regular variation conditions. The finite-sample coverage probabilities of our asymptotic confidence sets are evaluated in a simulation study for the half-space depth and the projection depth. We use the procedures in both chapters for risk management by applying them to stock market returns. The third essay develops a statistical inference theory of a recently proposed tail risk measure. For regulators who are interested in monitoring these what-we-called relative risks of individual banks, we provide a jackknife empirical likelihood inference procedure based on the smoothed nonparametric estimation and a Wilks type of result. A simulation study and real-life data analysis show that the proposed relative risk measure is useful in monitoring systemic risk.
|Qualification||Doctor of Philosophy|
|Award date||6 Dec 2016|
|Place of Publication||Tilburg|
|Publication status||Published - 2016|