Centrality and extremity in multivariate statistics: Data depth, extreme value theory, and applications

Multivariate statistics plays an important role in the modern era of information technology. Most existing multivariate analysis relies heavily on the assumption of normality. However, this normality assumption is often difficult to justify in practice, and thus nonparametric methodologies are needed. Much of the multivariate inference and applications evolve around the centrality and/or extremity of the data sets or their underlying distributions. In this dissertation we develop new nonparametric statistical methodologies by studying centrality and extremity of data, and demonstrate the usefulness of these methodologies in many real life applications. Specifically: (1) We use the concept of data depth to develop nonparametric inferences related to the centrality of multivariate distributions. These include: (i) constructing multivariate tolerance regions using multivariate spacings derived from data depth. The proposed tolerance region is a connected "central" region of the data which reflects the underlying probabilistic geometry, and thus has a more natural interpretation and implementation. Moreover, the construction of this tolerance region is completely nonparametric and thus has broad applicability, in especially multivariate quality control. (ii) developing tests for detecting location and scale differences in two multivariate samples by identifying particular changes in the centrality pattern of the pooled sample with respect to the two individual ones. The patterns of changes can be observed through the so-called DD-plots (depth vs depth plots). The proposed tests are nonparametric and can be even moment-free. (2) Motivated by two aviation safety projects sponsored by the FAA, we use the extreme value theory to establish a threshold system in terms of multivariate extreme quantiles for simultaneous monitoring of multiple risk indicators.;Finally, we propose to apply the extreme value theory to improve the estimate of data depth outside the convex hull of the data set. This provides an important linkage between data depth, which is useful for inference on centrality, and extreme value theory, which is useful for inference on extremity. The proposed estimate of sample data depth can improve the performance of data depth in applications where potential outlying probability masses are the focus points.