Depths for Scatter and Shape Parameters

Abstract

For scatter matrices, the introduced concept, coined halfspace scatter depth, extends the one from Chen, Gao and Ren (2018) to the non-centered case, and is in the same spirit as the one in Zhang (2002). Two concepts of depth are presented for shape parameters. The first one can be seen as the companion concept to halfspace scatter depth. The second one, of a sign nature, is the depth-based counterpart of the Tyler's M-estimate of shape.

Rather than focusing, as in earlier works, on deepest scatter and shape matrices, I will investigate the properties of the proposed depths and of the corresponding depth regions. This will be done under minimal assumptions. Interestingly, fully understanding these depth functions will require considering different geometries/topologies on the space of (scatter or shape) matrices. I will also discuss, in the spirit of Zuo and Serfling (2000), the structural properties a scatter depth should satisfy, and investigate whether or not these are met by the proposed depths. Finally, I will show how these depths can be used in detecting outliers in time series financial data.

This talk is based on the two papers Paindaveine and Van Bever (2018) & Paindaveine and Van Bever (2019).