Statistical depth meets computational geometry: a short survey <s> Depth notions <s> Statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data. Here we introduce several general structures for depth functions, classify many existing examples as special cases, and establish results on the possession, or lack thereof, of four key properties desirable for depth functions in general. Roughly speaking, these properties may be described as: affine invariance, maximality at center, monotonicity relative to deepest point, and vanishing at infinity. This provides a more systematic basis for selection of a depth function. In particular, from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors. <s> BIB001 </s> Statistical depth meets computational geometry: a short survey <s> Depth notions <s> In 1975 John Tukey proposed a multivariate median which is the 'deepest' point in a given data cloud in R^d. Later, in measuring the depth of an arbitrary point z with respect to the data, David Donoho and Miriam Gasko considered hyperplanes through z and determined its 'depth' by the smallest portion of data that are separated by such a hyperplane. Since then, these ideas has proved extremely fruitful. A rich statistical methodology has developed that is based on data depth and, more general, nonparametric depth statistics. General notions of data depth have been introduced as well as many special ones. These notions vary regarding their computability and robustness and their sensitivity to reflect asymmetric shapes of the data. According to their different properties they fit to particular applications. The upper level sets of a depth statistic provide a family of set-valued statistics, named depth-trimmed or central regions. They describe the distribution regarding its location, scale 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 on R^d, thus allowing for laws of large numbers and consistency results. It has also been extended from d-variate data to data in functional spaces. <s> BIB002
A data set consisting of n univariate points is usually ranked in ascending or descending order. Univariate order statistics (i.e., the 'kth smallest value out of n') and derived quantities have been studied extensively. The median is defined as the order statistic of rank (n + 1)/2 when n is odd, and as the average of the order statistics of ranks n/2 and (n + 2)/2 when n is even. The median and any other order statistic of a univariate data set can be computed in O(n) time. Generalization to higher dimensions is, however, not straightforward. Alternatively, univariate points may be ranked from the outside inward by assigning the most extreme data points depth 1, the second smallest and second largest data points depth 2, etc. The deepest point then equals the usual median of the sample. The advantage of this type of ranking is that it can be extended to higher dimensions more easily. This section gives an overview of several possible generalizations of depth and the median to multivariate settings. Surveys of statistical applications of multivariate data depth may be found in  , BIB001 , and BIB002 .