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SUMMARY:Model and estimator selection for density estimation with L2-loss 
 - Lucien Birgé\, Paris VI
DTSTART:20120612T150000Z
DTEND:20120612T160000Z
UID:TALK37972@talks.cam.ac.uk
CONTACT:Richard Samworth
DESCRIPTION:We consider here estimation of an unknown probability density 
 s belonging to L2(mu)\nwhere mu is a probability measure. We have at hand 
 n i.i.d. observations with density\ns and use the squared L2-norm as our l
 oss function.\nMuch has been proved about the risk of various types of est
 imators (projection estimators\, kernel estimators\, estimators based on m
 odel selection\, etc.) when s is\na bounded density with a known L_{\\inft
 y}-norm \\|s\\|_{\\infty}\, in which case risk bounds often depend on \\|s
 \\|_{\\infty}\, with a few exceptions like estimation using regular histog
 rams or estimation of densities belonging to some specific Besov spaces as
  shown by Raynaud-Bouret\, Rivoirard and Tuleau-Malot (2011). Here we do n
 ot want to put any restriction on s\, therefore considering also unbounded
  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 ad
 aptive estimation and estimator selection. When s \\in L_{\\infty} but \\|
 s\\|_{\\infty} is unknown\, we recover the results corresponding to a know
 n value of \\|s\\|_{\\infty}. Although of a purely theoretical nature (the
  resulting estimator cannot be explicitly computed)\, our method neverthel
 ess leads\nto results that are presently not reachable by more concrete me
 thods.
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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