Concepts of outlyingness for various data structures

The term "outlier" is probably one of the vaguest and most imprecise ones in statistical science. There is no formal definition of an outlier, which all statisticians agree upon. However, for a univariate normal null-model Davies and Gather ([12] [13]) have introduced the concept of a-outliers and a-outlier regions, giving a definition which characterizes outliers only by their location relative to the assumed model for the good data. Outliers are thereby data points, observed in a region of the support of the anticipated distribution, namely an a-outlier region, where observations are - in a certain sense - unlikely under the assumed model. In this chapter we revisit this approach to outlyingness and generalize it to a variety of univariate and multivariate, continuous and discrete distributions as well as to structured models such as regression models and contingency tables. We also indicate how the concept of outlier regions can be used to define and construct outlier identification procedures