Robert Serfling
University of Texas at Dallas

Invariance and Equivariance: Benefits, Costs, and Methods

With univariate data, the branding of points as “outliers” remains invariant under linear transformation, while the median and other quantiles transform equivariantly. With multivariate data, we consider affine transformations and desire such properties to hold for various notions of multivariate outlyingness and quantile functions as well as other kinds of statistics. Further, striking geometric features or structures perceived in a data set should be affinely invariant or else discarded as mere artifacts of the particular coordinate system. However, these invariance or equivariance properties do not always hold as may be desired. This talk treats several approaches toward resolution of any such shortcoming. One approach involves measuring the shortfall. More recent approaches enable such properties to hold through preprocessing of the data by either a “transformation-retransformation” (TR) transformation or an “invariant coordinate system” (ICS) transformation. Such TR and ICS transformations are examined closely within a coherent framework. Their formulation, sample constructions, theoretical properties, practical applications, computational aspects, and some challenging open issues are reviewed. Also, the costs in terms of trade-offs against optimality, robustness, efficiency, and computational ease are discussed.

Key words and phrases: Equivariance and invariance; Robustness; Multivariate; Computation; Dimensionality; Efficiency; Nonparametric.