On Obtaining Affine Equivariance or Invariance of Multivariate Statistics, with Application to Quantile and Outlyingness Functions

• Activity Number: 558
• Type: Invited
• Date/Time: Wednesday, August 4, 2010 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Robert Serfling*+
• Companies: The University of Texas at Dallas
• Keywords: Equivariance and invariance ; Multivariate ; Nonparametric ; Outlier identification

Abstract

Equivariance and invariance issues are fundamental but often problematic aspects of multivariate statistical analysis. Some general transformation techniques are discussed which applied to data can produce affine equivariant or invariant statistics from versions lacking the desired property. In particular, "strong invariant coordinate system" (SICS) transformations are introduced and studied. It is seen that geometric artifacts of SICS-standardized data are invariant under affine transformation of the original data followed by standardization using the same SICS functional, subject only to translation and homogeneous scale change. Applications are discussed, including formulation of competitive new affine invariant multivariate outlyingness functions and modification of existing ones lacking affine invariance to versions possessing it.