73 research outputs found
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
Optimal change-point estimation from indirect observations
We study nonparametric change-point estimation from indirect noisy
observations. Focusing on the white noise convolution model, we consider two
classes of functions that are smooth apart from the change-point. We establish
lower bounds on the minimax risk in estimating the change-point and develop
rate optimal estimation procedures. The results demonstrate that the best
achievable rates of convergence are determined both by smoothness of the
function away from the change-point and by the degree of ill-posedness of the
convolution operator. Optimality is obtained by introducing a new technique
that involves, as a key element, detection of zero crossings of an estimate of
the properly smoothed second derivative of the underlying function.Comment: Published at http://dx.doi.org/10.1214/009053605000000750 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On deconvolution of distribution functions
The subject of this paper is the problem of nonparametric estimation of a
continuous distribution function from observations with measurement errors. We
study minimax complexity of this problem when unknown distribution has a
density belonging to the Sobolev class, and the error density is ordinary
smooth. We develop rate optimal estimators based on direct inversion of
empirical characteristic function. We also derive minimax affine estimators of
the distribution function which are given by an explicit convex optimization
problem. Adaptive versions of these estimators are proposed, and some numerical
results demonstrating good practical behavior of the developed procedures are
presented.Comment: Published in at http://dx.doi.org/10.1214/11-AOS907 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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