Institute of Mathematical Statistics Lecture Notes - Monograph Series

Some facts about functionals of location and scatter

R. M. Dudley

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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

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

First available in Project Euclid: 28 November 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

equivariance $t$ distributions

Copyright © 2006, Institute of Mathematical Statistics


Dudley, R. M. Some facts about functionals of location and scatter. High Dimensional Probability, 207--219, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000860.

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