Some facts about functionals of location and scatter

Details Some facts about functionals of location and scatter Some facts about functionals of location and scatter

2006-12-22

arxiv.org/abs/math/0612709v1

IMS Lecture Notes Monograph Series 2006, Vol. 51, 207-219

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/074921706000000860 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/p

Scientific paper

10.1214/074921706000000860

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 ${\Bbb{R}}^d$ 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 ${\Bbb{R}}^d$, following up on work of Kent, Tyler, and D\"{u}mbgen.

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