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B0692
Title: Aspect-based depth statistics for functional data Authors:  Karl Mosler - Universitaet zu Koeln (Germany) [presenting]
Abstract: A depth statistic measures how central an observation is located in other data or, more general, in a given probability distribution. Various depth notions have been proposed for multivariate data and successfully applied to problems in data analysis and non-parametric statistics. Recently, depth statistics have been extended to the analysis of functional data. If the data are in $R^d$, a depth statistic is usually assumed to be affine-invariant (or invariant to translation and scale), monotonously decreasing from a point of maximum depth, vanishing at infinity, upper semicontinuous, and quasi-concave. While these postulates are well established for $d$-variate data, it is not obvious how to extend them to data in more general linear spaces, viz. spaces of functions. It is argued that the statistical analysis of functional data is led by certain aspects in which the analyst is interested, and discusses how these aspects enter the notion in a formalized way. A general aspect-based approach is introduced for the definition of depths for functional data and investigates sets of postulates for them. The aspects correspond to linear functionals on the function spaces. Two principal ways are considered to aggregate centrality information over the time axis: infimum depth and mean depth, where the latter includes a weight term. Known functional depths appear as special cases.