|On adaptive minimax density estimation on R^d |
Auteur(s): Goldenshluger Alexander, Lepski O.
(Article) Publié: Probability Theory And Related Fields, vol. 159 p.479–543. (2014)
Ref HAL: hal-01265245_v1
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We address the problem of adaptive minimax density estimation on $\bR^d$with $\bL_p$--loss on the anisotropic Nikol'skii classes.We fully characterize behavior of the minimax risk for differentrelationships between regularity parameters andnorm indexes in definitions of the functional class and of the risk.In particular, we show that there are fourdifferent regimes with respectto the behavior of the minimax risk.We developa single estimator which is (nearly) optimal in orderover the complete scale of the anisotropic Nikol'skii classes.Our estimation procedure is basedon a data-driven selection of an estimator from a fixedfamily ofkernel estimators.