A review of Tyler's shape matrix and its extensions

: Taskinen, Sara; Frahm, Gabriel; Nordhausen, Klaus; Oja, Hannu

: Yi, Mengxi; Nordhausen, Klaus

: 1

Publisher: Springer International Publishing

: 2023

: Robust and Multivariate Statistical Methods. Festschrift in Honor of David E. Tyler

: Robust and Multivariate Statistical Methods: Festschrift in Honor of David E. Tyler

: 23

: 41

: 978-3-031-22686-1

: 978-3-031-22687-8

DOI: https://doi.org/10.1007/978-3-031-22687-8_2

: https://link.springer.com/chapter/10.1007/978-3-031-22687-8_2

In a seminal paper, Tyler (1987a) suggests an M-estimator for shape, which is now known as Tyler's shape matrix. Tyler's shape matrix is increasingly popular due to its nice statistical properties. It is distribution free within the class of generalized elliptical distributions. Further, under very mild regularity conditions, it is consistent and asymptotically normally distributed after the usual standardization. Tyler's shape matrix is still the subject of active research, e.g., in the signal processing literature, which discusses structured and regularized shape matrices. In this article, we review Tyler's original shape matrix and some recent developments.