Linear and convex aggregation of density estimators |
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Authors: | Ph Rigollet A B Tsybakov |
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Institution: | (1) Georgia Institute of Technology, USA;(2) Université Paris — VI, France |
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Abstract: | We study the problem of finding the best linear and convex combination of M estimators of a density with respect to the mean squared risk. We suggest aggregation procedures and we prove sharp oracle
inequalities for their risks, i.e., oracle inequalities with leading constant 1. We also obtain lower bounds showing that
these procedures attain optimal rates of aggregation. As an example, we consider aggregation of multivariate kernel density
estimators with different bandwidths. We show that linear and convex aggregates mimic the kernel oracles in asymptotically
exact sense. We prove that, for Pinsker’s kernel, the proposed aggregates are sharp asymptotically minimax simultaneously
over a large scale of Sobolev classes of densities. Finally, we provide simulations demonstrating performance of the convex
aggregation procedure.
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Keywords: | aggregation oracle inequalities statistical learning nonparametric density estimation sharp minimax adaptivity kernel estimates of a density |
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