2,299 research outputs found
Nonlinear estimation for linear inverse problems with error in the operator
We study two nonlinear methods for statistical linear inverse problems when
the operator is not known. The two constructions combine Galerkin
regularization and wavelet thresholding. Their performances depend on the
underlying structure of the operator, quantified by an index of sparsity. We
prove their rate-optimality and adaptivity properties over Besov classes.Comment: Published in at http://dx.doi.org/10.1214/009053607000000721 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On adaptive inference and confidence bands
The problem of existence of adaptive confidence bands for an unknown density
that belongs to a nested scale of H\"{o}lder classes over or
is considered. Whereas honest adaptive inference in this problem is
impossible already for a pair of H\"{o}lder balls ,
of fixed radius, a nonparametric distinguishability condition is introduced
under which adaptive confidence bands can be shown to exist. It is further
shown that this condition is necessary and sufficient for the existence of
honest asymptotic confidence bands, and that it is strictly weaker than similar
analytic conditions recently employed in Gin\'{e} and Nickl [Ann. Statist. 38
(2010) 1122--1170]. The exceptional sets for which honest inference is not
possible have vanishingly small probability under natural priors on H\"{o}lder
balls . If no upper bound for the radius of the H\"{o}lder balls is
known, a price for adaptation has to be paid, and near-optimal adaptation is
possible for standard procedures. The implications of these findings for a
general theory of adaptive inference are discussed.Comment: Published in at http://dx.doi.org/10.1214/11-AOS903 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Nonparametric estimation of scalar diffusions based on low frequency data
We study the problem of estimating the coefficients of a diffusion (X_t,t\geq
0); the estimation is based on discrete data X_{n\Delta},n=0,1,...,N. The
sampling frequency \Delta^{-1} is constant, and asymptotics are taken as the
number N of observations tends to infinity. We prove that the problem of
estimating both the diffusion coefficient (the volatility) and the drift in a
nonparametric setting is ill-posed: the minimax rates of convergence for
Sobolev constraints and squared-error loss coincide with that of a,
respectively, first- and second-order linear inverse problem. To ensure
ergodicity and limit technical difficulties we restrict ourselves to scalar
diffusions living on a compact interval with reflecting boundary conditions.
Our approach is based on the spectral analysis of the associated Markov
semigroup. A rate-optimal estimation of the coefficients is obtained via the
nonparametric estimation of an eigenvalue-eigenfunction pair of the transition
operator of the discrete time Markov chain (X_{n\Delta},n=0,1,...,N) in a
suitable Sobolev norm, together with an estimation of its invariant density.Comment: Published at http://dx.doi.org/10.1214/009053604000000797 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Early stopping for statistical inverse problems via truncated SVD estimation
We consider truncated SVD (or spectral cut-off, projection) estimators for a
prototypical statistical inverse problem in dimension . Since calculating
the singular value decomposition (SVD) only for the largest singular values is
much less costly than the full SVD, our aim is to select a data-driven
truncation level only based on the knowledge of
the first singular values and vectors. We analyse in detail
whether sequential {\it early stopping} rules of this type can preserve
statistical optimality. Information-constrained lower bounds and matching upper
bounds for a residual based stopping rule are provided, which give a clear
picture in which situation optimal sequential adaptation is feasible. Finally,
a hybrid two-step approach is proposed which allows for classical oracle
inequalities while considerably reducing numerical complexity.Comment: slightly modified version. arXiv admin note: text overlap with
arXiv:1606.0770
On adaptive posterior concentration rates
We investigate the problem of deriving posterior concentration rates under
different loss functions in nonparametric Bayes. We first provide a lower bound
on posterior coverages of shrinking neighbourhoods that relates the metric or
loss under which the shrinking neighbourhood is considered, and an intrinsic
pre-metric linked to frequentist separation rates. In the Gaussian white noise
model, we construct feasible priors based on a spike and slab procedure
reminiscent of wavelet thresholding that achieve adaptive rates of contraction
under or metrics when the underlying parameter belongs to a
collection of H\"{o}lder balls and that moreover achieve our lower bound. We
analyse the consequences in terms of asymptotic behaviour of posterior credible
balls as well as frequentist minimax adaptive estimation. Our results are
appended with an upper bound for the contraction rate under an arbitrary loss
in a generic regular experiment. The upper bound is attained for certain sieve
priors and enables to extend our results to density estimation.Comment: Published at http://dx.doi.org/10.1214/15-AOS1341 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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