374 research outputs found
Universal pointwise selection rule in multivariate function estimation
In this paper, we study the problem of pointwise estimation of a multivariate
function. We develop a general pointwise estimation procedure that is based on
selection of estimators from a large parameterized collection. An upper bound
on the pointwise risk is established and it is shown that the proposed
selection procedure specialized for different collections of estimators leads
to minimax and adaptive minimax estimators in various settings.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ144 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Estimation in the convolution structure density model. Part I: oracle inequalities
We study the problem of nonparametric estimation under \bL_p-loss, , in the framework of the convolution structure density model on
\bR^d. This observation scheme is a generalization of two classical
statistical models, namely density estimation under direct and indirect
observations. In Part I the original pointwise selection rule from a family of
"kernel-type" estimators is proposed. For the selected estimator, we prove an
\bL_p-norm oracle inequality and several of its consequences. In Part II the
problem of adaptive minimax estimation under \bL_p--loss over the scale of
anisotropic Nikol'skii classes is addressed. We fully characterize the behavior
of the minimax risk for different relationships between regularity parameters
and norm indexes in the definitions of the functional class and of the risk. We
prove that the selection rule proposed in Part I leads to the construction of
an optimally or nearly optimally (up to logarithmic factor) adaptive estimator
On adaptive minimax density estimation on
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 different relationships between regularity
parameters and norm indexes in definitions of the functional class and of the
risk. In particular, we show that there are four different regimes with respect
to the behavior of the minimax risk. We develop a single estimator which is
(nearly) optimal in orderover the complete scale of the anisotropic Nikol'skii
classes. Our estimation procedure is based on a data-driven selection of an
estimator from a fixed family of kernel estimators
Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality
We address the problem of density estimation with -loss by
selection of kernel estimators. We develop a selection procedure and derive
corresponding -risk oracle inequalities. It is shown that the
proposed selection rule leads to the estimator being minimax adaptive over a
scale of the anisotropic Nikol'skii classes. The main technical tools used in
our derivations are uniform bounds on the -norms of empirical
processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011),
to appear].Comment: Published in at http://dx.doi.org/10.1214/11-AOS883 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Estimation in the convolution structure density model. Part II: adaptation over the scale of anisotropic classes
This paper continues the research started in \cite{LW16}. In the framework of
the convolution structure density model on \bR^d, we address the problem of
adaptive minimax estimation with \bL_p--loss over the scale of anisotropic
Nikol'skii classes. We fully characterize the behavior of the minimax risk for
different relationships between regularity parameters and norm indexes in the
definitions of the functional class and of the risk. In particular, we show
that the boundedness of the function to be estimated leads to an essential
improvement of the asymptotic of the minimax risk. We prove that the selection
rule proposed in Part I leads to the construction of an optimally or nearly
optimally (up to logarithmic factor) adaptive estimator
Upper functions for positive random functionals. Application to the empirical processes theory II
International audienceThis part of the paper finalizes the research started in Lepski (2013b)
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