Projection-based depth functions and associated medians

A half-region depth for functional data

“…This functional depth provides a criterion for ordering the sample of curves from center-outward. The notion of statistical depth was first analyzed for multivariate observations, and many different definitions of depth have been studied in the literature, for example Mahalanobis (1936), Tukey (1975), Oja (1983), Liu (1990), Donoho and Gasko (1992), Liu et al (1999), Zuo and Serfling (2000) and Zuo (2003). Most of these multivariate depths are not adequate for high-dimensional data, therefore their applicability is restricted to low-dimensional vector observations.…”

Comparing quantile residual life functions by confidence bands

“…Several depth definitions for multivariate data have been proposed and analyzed by Mahalanobis (1936), Tukey (1975), Oja (1983), Liu (1990), Singh (1991), Fraiman and Meloche (1999), Vardi and Zhang (2000), Koshevoy and Mosler (1997) and Zuo (2003). Direct generalization of current multivariate depths to functional data often leads to either depths that are computationally intractable or depths that do not take into account some natural properties of the functions, such as shape.…”

On Exact Computation of Some Statistics Based on Projection Pursuit in a General Regression Context

“…Let us generalize location projection depth, thoroughly investigated in Zuo (2003), to the points in the general regression setup. In the empirical case discussed here, the projection regression depth P D n (y, x) of y ∈ R m given x = (1, z ) ∈ 1 × R p may be defined as…”

“…The projection pursuit plays a crucial role in the definitions of various statistical depths including halfspace depth, weighted halfspace depth and projection depth; see Rousseeuw and Ruts (1999), Hlubinka et al (2010) and Zuo (2003) with references therein, respectively.…”