Depth Statistics

By Karl Mosler. Encyclopedia of Image Processing, 2012

121 citations recorded by the source corpus.

In 1975 John Tukey proposed a multivariate median which is th e ‘deepest’ point in a given data cloud inRd (Tukey, 1975). In measuring the depth of an arbitrary point z with respect to the data, Donoho and Gasko (1992) considered hyperplanes through z and determined its ‘depth’ by the smallest portion of data th at are separated by such a hyperplane. Since then, this idea has proved extremel y fruitful. A rich statistical methodology has developed that is based on data dep th and, more general, nonparametric depth statistics. General notions of data de pth have been introduced as well as many special ones. These notions vary regarding th eir computability and robustness and their sensitivity to reflect asymmetric shap es of the data. According to their different properties they fit to particular applica t ons. The upper level sets of a depth statistic provide a family of set-valued statisti cs, nameddepth-trimmed or central regions. They describe the distribution regarding its location, sc ale and shape. The most central region serves as a median. The notion of depth has been extended from data clouds, that is empirical distributions , to general probability distributions onRd, thus allowing for laws of large numbers and consistency res ults. It has also been extended from d-variate data to data in functional spaces. The present chapter surveys the theory and methodology of depth statist ics. Recent reviews on data depth are given in Cascos (2009) and Se rfling (2006). Liu et al. (2006) collects theoretical as well as applied wor k. More on the theory of depth functions and many details are found in Zuo and Serfli ng (2000) and the monograph by Mosler (2002). The depth of a data point is reversely related to its outlyingness , and the depthtrimmed regions can be seen as multivariate set-valued quantiles. To illustrate the notions we consider bivariate data from the EU-27 countries regarding unemploy-

Subfields: Artificial Intelligence, History and Philosophy of Science, Statistics and Probability

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