## Institute of Mathematical Statistics Lecture Notes - Monograph Series

### Some facts about functionals of location and scatter

R. M. Dudley

#### Abstract

Assumptions on a likelihood function, including a local Glivenko-Cantelli condition, imply the existence of M-estimators converging to an M-functional. Scatter matrix-valued estimators, defined on all empirical measures on $\RR^d$ for $d\geq 2$, and equivariant under all, including singular, affine transformations, are shown to be constants times the sample covariance matrix. So, if weakly continuous, they must be identically 0. Results are stated on existence and differentiability of location and scatter functionals, defined on a weakly dense, weakly open set of laws, via elliptically symmetric t distributions on $\RR^d$, following up on work of Kent, Tyler, and Dümbgen.

#### Chapter information

Source
Evarist Giné, Vladimir Koltchinskii, Wenbo Li, Joel Zinn, eds., High Dimensional Probability: Proceedings of the Fourth International Conference (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006), 207-219

Dates
First available in Project Euclid: 28 November 2007

https://projecteuclid.org/euclid.lnms/1196284114

Digital Object Identifier
doi:10.1214/074921706000000860

Mathematical Reviews number (MathSciNet)
MR2387771

Zentralblatt MATH identifier
1117.62056

Subjects
Primary: 62G05: Estimation 62GH20
Secondary: 62G35: Robustness

Keywords
equivariance $t$ distributions

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