On Depth and Deep Points: A Calculus

For a general definition of depth in data analysis a differential-like calculus is constructed in which the location case (the framework of Tukey's median) plays a fundamental role similar to that of linear functions in the mathematical analysis. As an application, a lower bound for maximal reg...

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Bibliographic Details
Published in:The Annals of statistics 2002-12, Vol.30 (6), p.1681-1736
Main Author: Mizera, Ivan
Format: Article
Language:English
Subjects:
52A40
54C60
55M25
62H05
90C29
Centerpoint
Compactification
degree of mapping
depth
Estimation and Robustness
Estimators
Exact sciences and technology
General topics
General topology
halfspace
Hemispheres
Kronecker index
Linear inference, regression
Linear regression
Mathematical theorems
Mathematical vectors
Mathematics
median
Multivariate analysis
multivariate location
Probability and statistics
Random variables
regression
Sciences and techniques of general use
set-valued analysis
Statistics
Tangents
Topological spaces
Topological theorems
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
vector optimization
weak convergence
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Description
Summary:For a general definition of depth in data analysis a differential-like calculus is constructed in which the location case (the framework of Tukey's median) plays a fundamental role similar to that of linear functions in the mathematical analysis. As an application, a lower bound for maximal regression depth is proved in the general multidimensional case-as conjectured by Rousseeuw and Hubert and others. This lower bound is demonstrated to have an impact on the breakdown point of the maximum depth estimator.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/1043351254