Non-parametric Statistical Methods - Applications in MALDI Imaging and Finance

Abstract

This thesis contains applications of (non-)parametric statistical methods (and the development of such techniques) with a focus on applications to three distinct topics:

1) Computational statistics, specifically the (efficient and exact) calculation of the joint distribution of order statistics. Since ranks are fundamental to many statistical methods, these have many applications, some of which are detailed. 2) Mathematical finance, specifically results on "reverse stress testing" which, roughly speaking, has the goal of performing a data-driven selection of likely scenarios for which a given portfolio exceeds a specified loss. Two notable contributions are the development of non-parametric confidence regions in elliptical models and a characterisation of the subspace which, in skew-elliptical models, contains the sought scenario. 3) Mathematical statistics, specifically methods with applications to the statistical analysis of biomedical images. One focus is on statistical tests based on correlation coefficients when one of the random variables is a binary random variable. The derived results are utilised to elucidate some statistical properties of matrix-assisted laser desorption/ionization (MALDI) mass spectroscopy data.

In my work on all of these topics, my focus was on developing and applying statistical methods that are based only on the absolutely necessary assumptions. This is, of course, an aspirational goal. I am, however hopeful that I was able to make my own small contribution to the science of mathematical statistics.