941 research outputs found
Asymptotic equivalence for regression under fractional noise
Consider estimation of the regression function based on a model with
equidistant design and measurement errors generated from a fractional Gaussian
noise process. In previous literature, this model has been heuristically linked
to an experiment, where the anti-derivative of the regression function is
continuously observed under additive perturbation by a fractional Brownian
motion. Based on a reformulation of the problem using reproducing kernel
Hilbert spaces, we derive abstract approximation conditions on function spaces
under which asymptotic equivalence between these models can be established and
show that the conditions are satisfied for certain Sobolev balls exceeding some
minimal smoothness. Furthermore, we construct a sequence space representation
and provide necessary conditions for asymptotic equivalence to hold.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1262 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Nonparametric regression using deep neural networks with ReLU activation function
Consider the multivariate nonparametric regression model. It is shown that
estimators based on sparsely connected deep neural networks with ReLU
activation function and properly chosen network architecture achieve the
minimax rates of convergence (up to -factors) under a general
composition assumption on the regression function. The framework includes many
well-studied structural constraints such as (generalized) additive models.
While there is a lot of flexibility in the network architecture, the tuning
parameter is the sparsity of the network. Specifically, we consider large
networks with number of potential network parameters exceeding the sample size.
The analysis gives some insights into why multilayer feedforward neural
networks perform well in practice. Interestingly, for ReLU activation function
the depth (number of layers) of the neural network architectures plays an
important role and our theory suggests that for nonparametric regression,
scaling the network depth with the sample size is natural. It is also shown
that under the composition assumption wavelet estimators can only achieve
suboptimal rates.Comment: article, rejoinder and supplementary materia
Posterior contraction rates for support boundary recovery
Given a sample of a Poisson point process with intensity we study recovery of the boundary function from a
nonparametric Bayes perspective. Because of the irregularity of this model, the
analysis is non-standard. We establish a general result for the posterior
contraction rate with respect to the -norm based on entropy and one-sided
small probability bounds. From this, specific posterior contraction results are
derived for Gaussian process priors and priors based on random wavelet series
Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise
We consider the models Y_{i,n}=\int_0^{i/n}
\sigma(s)dW_s+\tau(i/n)\epsilon_{i,n}, and \tilde
Y_{i,n}=\sigma(i/n)W_{i/n}+\tau(i/n)\epsilon_{i,n}, i=1,...,n, where W_t
denotes a standard Brownian motion and \epsilon_{i,n} are centered i.i.d.
random variables with E(\epsilon_{i,n}^2)=1 and finite fourth moment.
Furthermore, \sigma and \tau are unknown deterministic functions and W_t and
(\epsilon_{1,n},...,\epsilon_{n,n}) are assumed to be independent processes.
Based on a spectral decomposition of the covariance structures we derive series
estimators for \sigma^2 and \tau^2 and investigate their rate of convergence of
the MISE in dependence of their smoothness. To this end specific basis
functions and their corresponding Sobolev ellipsoids are introduced and we show
that our estimators are optimal in minimax sense. Our work is motivated by
microstructure noise models. Our major finding is that the microstructure noise
\epsilon_{i,n} introduces an additionally degree of ill-posedness of 1/2;
irrespectively of the tail behavior of \epsilon_{i,n}. The method is
illustrated by a small numerical study.Comment: 5 figures, corrected references, minor change
The Le Cam distance between density estimation, Poisson processes and Gaussian white noise
It is well-known that density estimation on the unit interval is
asymptotically equivalent to a Gaussian white noise experiment, provided the
densities have H\"older smoothness larger than and are uniformly bounded
away from zero. We derive matching lower and constructive upper bounds for the
Le Cam deficiencies between these experiments, with explicit dependence on both
the sample size and the size of the densities in the parameter space. As a
consequence, we derive sharp conditions on how small the densities can be for
asymptotic equivalence to hold. The related case of Poisson intensity
estimation is also treated.Comment: Some results from an earlier version of this preprint have been moved
to arXiv:1802.0342
Lower bounds for volatility estimation in microstructure noise models
In this paper we derive lower bounds in minimax sense for estimation of the
instantaneous volatility if the diffusion type part cannot be observed directly
but under some additional Gaussian noise. Three different models are
considered. Our technique is based on a general inequality for Kullback-Leibler
divergence of multivariate normal random variables and spectral analysis of the
processes. The derived lower bounds are indeed optimal. Upper bounds can be
found in Munk and Schmidt-Hieber [18]. Our major finding is that the Gaussian
microstructure noise introduces an additional degree of ill-posedness for each
model, respectively.Comment: 16 page
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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