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CATEGORIES:Statistics
SUMMARY:Model and estimator selection for density estimati
on with L2-loss - Lucien BirgĂ©\, Paris VI
DTSTART;TZID=Europe/London:20120612T160000
DTEND;TZID=Europe/London:20120612T170000
UID:TALK37972AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/37972
DESCRIPTION:We consider here estimation of an unknown probabil
ity density s belonging to L2(mu)\nwhere mu is a p
robability measure. We have at hand n i.i.d. obser
vations with density\ns and use the squared L2-nor
m as our loss function.\nMuch has been proved abou
t the risk of various types of estimators (project
ion estimators\, kernel estimators\, estimators ba
sed on model selection\, etc.) when s is\na bounde
d density with a known L_{\\infty}-norm \\|s\\|_{\
\infty}\, in which case risk bounds often depend o
n \\|s\\|_{\\infty}\, with a few exceptions like e
stimation using regular histograms or estimation o
f densities belonging to some specific Besov space
s as shown by Raynaud-Bouret\, Rivoirard and Tulea
u-Malot (2011). Here we do not want to put any res
triction on s\, therefore considering also unbound
ed densities or bounded densities with unknown L_{
\\infty}-norm.\n\nWe shall deal with estimation by
model selection\, allowing arbitrary families of
finite-dimensional (possibly non-linear) models\,
with applications to adaptive estimation and esti
mator selection. When s \\in L_{\\infty} but \\|s\
\|_{\\infty} is unknown\, we recover the results c
orresponding to a known value of \\|s\\|_{\\infty}
. Although of a purely theoretical nature (the res
ulting estimator cannot be explicitly computed)\,
our method nevertheless leads\nto results that are
presently not reachable by more concrete methods.
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:Richard Samworth
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