The Concept of α-Outliers in Structured Data Situations

In every statistical data analysis, somehow surprising observations can occur which deviate strongly from the remaining observations or the assumed model. On the one hand, these observations may contain important pieces of information about the data-generating process. On the other hand, they might simply be measurement or reporting errors. Regardless of which origin the observation has, it is commonly named “outlier”. There are numerous ways to detect outliers, with no strategy outperforming others in every situation. Besides non-parametric procedures, e.g., based on depth measures, also model-based strategies exist. In order to be able to detect outliers, it first needs to be specified what is meant by an outlier. In this contribution, we discuss the notion of α-outliers as introduced by Davies and Gather (1993). The basic idea is that there exists a pattern which is supported by the majority of the data. Observations which are strongly deviating from this pattern are understood as outliers. Within the α-outlier concept, the pattern is the statistical model one has in mind for the data generating mechanism. Observations which lie in a region with low probability and are thereby surprising are understood as outliers. The general idea of α-outliers can be applied to basically any statistical model. The so-called outlier region usually is uniquely defined for a given statistical distribution. However, within the analysis of observed data sets this is often only specified up to some unknown parameters of the assumed class of distributions, resulting in the necessity of outlier identification procedures. This chapter is structured as follows: Sect. 6.2 reviews the general definition of α-outlier regions. One-step approaches towards the detection of α-outliers in a