Thursday, February 25th, 2010 at 4.10pm, MSB 1147 (Colloquium Room)
Refreshments: 3.30pm, MSB 4110 (Statistics Lounge)
Speaker: Robert Serfling (U Texas, Dallas)
Title: Constructing Nonparametric Multivariate Outlyingness Functions that are Robust, Computationally Easy, Affine Invariant, and Efficient
Abstract: Identification of possible outliers in the data is of paramount importance. Here we consider the setting of nonparametric multivariate statistical procedures. Robust versions of Mahalanobis distance outlyingness functions are highly versatile and popular, being affine invariant and efficient, but, however, they impose ellipsoidal contours.
Free of this constraint is the spatial outlyingness, which, however, lacks full affine invariance, although a modified sample version possesses it.
Further, the spatial approach lacks sufficient robustness. An adequately robust alternative not requiring ellipsoidal contours is the projection outlyingness, but it is computationally intensive and somewhat inefficient. Can we develop outlyingness functions which retain the favorable properties of Mahalanobis distance without confining to ellipsoidal contours? The first part of the talk provides a brief overview of multivariate outlyingness functions (in passing noting connections to multivariate depth, quantile, and rank functions). The second part introduces three new tools which may be applied to address the above issues. One is a result that following standardization of the data using any of a certain wide class of transformations, the spatial outlyingness function becomes affine invariant. A second result introduces a certain narrower class of standardizing transformations such that many other statistical procedures acquire full invariance or equivariance properties as desired, when carried out on the standardized data. A third result provides a method of robustification of the spatial outlyingness function.
The final part of the talk treats application of these results, in conjunction with both the spatial approach and the projection pursuit approach, to construct some new outlyingness functions that are effective competitors to Mahalanobis distance outlyingness with some improvements when ellipsoidal contours are not apropos. Other applications also will be noted.
