33,479 research outputs found
Robust nonparametric estimation via wavelet median regression
In this paper we develop a nonparametric regression method that is
simultaneously adaptive over a wide range of function classes for the
regression function and robust over a large collection of error distributions,
including those that are heavy-tailed, and may not even possess variances or
means. Our approach is to first use local medians to turn the problem of
nonparametric regression with unknown noise distribution into a standard
Gaussian regression problem and then apply a wavelet block thresholding
procedure to construct an estimator of the regression function. It is shown
that the estimator simultaneously attains the optimal rate of convergence over
a wide range of the Besov classes, without prior knowledge of the smoothness of
the underlying functions or prior knowledge of the error distribution. The
estimator also automatically adapts to the local smoothness of the underlying
function, and attains the local adaptive minimax rate for estimating functions
at a point. A key technical result in our development is a quantile coupling
theorem which gives a tight bound for the quantile coupling between the sample
medians and a normal variable. This median coupling inequality may be of
independent interest.Comment: Published in at http://dx.doi.org/10.1214/07-AOS513 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Nonparametric regression in exponential families
Most results in nonparametric regression theory are developed only for the
case of additive noise. In such a setting many smoothing techniques including
wavelet thresholding methods have been developed and shown to be highly
adaptive. In this paper we consider nonparametric regression in exponential
families with the main focus on the natural exponential families with a
quadratic variance function, which include, for example, Poisson regression,
binomial regression and gamma regression. We propose a unified approach of
using a mean-matching variance stabilizing transformation to turn the
relatively complicated problem of nonparametric regression in exponential
families into a standard homoscedastic Gaussian regression problem. Then in
principle any good nonparametric Gaussian regression procedure can be applied
to the transformed data. To illustrate our general methodology, in this paper
we use wavelet block thresholding to construct the final estimators of the
regression function. The procedures are easily implementable. Both theoretical
and numerical properties of the estimators are investigated. The estimators are
shown to enjoy a high degree of adaptivity and spatial adaptivity with
near-optimal asymptotic performance over a wide range of Besov spaces. The
estimators also perform well numerically.Comment: Published in at http://dx.doi.org/10.1214/09-AOS762 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Effect of mean on variance function estimation in nonparametric regression
Variance function estimation in nonparametric regression is considered and
the minimax rate of convergence is derived. We are particularly interested in
the effect of the unknown mean on the estimation of the variance function. Our
results indicate that, contrary to the common practice, it is not desirable to
base the estimator of the variance function on the residuals from an optimal
estimator of the mean when the mean function is not smooth. Instead it is more
desirable to use estimators of the mean with minimal bias. On the other hand,
when the mean function is very smooth, our numerical results show that the
residual-based method performs better, but not substantial better than the
first-order-difference-based estimator. In addition our asymptotic results also
correct the optimal rate claimed in Hall and Carroll [J. Roy. Statist. Soc.
Ser. B 51 (1989) 3--14].Comment: Published in at http://dx.doi.org/10.1214/009053607000000901 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The thermodynamic evolution of the cosmological event horizon
By manipulating the integral expression for the proper radius of the
cosmological event horizon (CEH) in a Friedmann-Robertson-Walker (FRW)
universe, we obtain an analytical expression for the change \dd R_e in
response to a uniform fluctuation \dd\rho in the average cosmic background
density . We stipulate that the fluctuation arises within a vanishing
interval of proper time, during which the CEH is approximately stationary, and
evolves subsequently such that \dd\rho/\rho is constant. The respective
variations 2\pi R_e \dd R_e and \dd E_e in the horizon entropy and
enclosed energy should be therefore related through the cosmological
Clausius relation. In that manner we find that the temperature of the CEH
at an arbitrary time in a flat FRW universe is , which recovers
asymptotically the usual static de Sitter temperature. Furthermore, it is
proven that during radiation-dominance and in late times the CEH conforms to
the fully dynamical First Law T_e \drv S_e = P\drv V_e - \drv E_e, where
is the enclosed volume and is the average cosmic pressure.Comment: 6 page
Delay-Coordinates Embeddings as a Data Mining Tool for Denoising Speech Signals
In this paper we utilize techniques from the theory of non-linear dynamical
systems to define a notion of embedding threshold estimators. More specifically
we use delay-coordinates embeddings of sets of coefficients of the measured
signal (in some chosen frame) as a data mining tool to separate structures that
are likely to be generated by signals belonging to some predetermined data set.
We describe a particular variation of the embedding threshold estimator
implemented in a windowed Fourier frame, and we apply it to speech signals
heavily corrupted with the addition of several types of white noise. Our
experimental work seems to suggest that, after training on the data sets of
interest,these estimators perform well for a variety of white noise processes
and noise intensity levels. The method is compared, for the case of Gaussian
white noise, to a block thresholding estimator
Breathing oscillations of a trapped impurity in a Bose gas
Motivated by a recent experiment [J. Catani et al., arXiv:1106.0828v1
preprint, 2011], we study breathing oscillations in the width of a harmonically
trapped impurity interacting with a separately trapped Bose gas. We provide an
intuitive physical picture of such dynamics at zero temperature, using a
time-dependent variational approach. In the Gross-Pitaevskii regime we obtain
breathing oscillations whose amplitudes are suppressed by self trapping, due to
interactions with the Bose gas. Introducing phonons in the Bose gas leads to
the damping of breathing oscillations and non-Markovian dynamics of the width
of the impurity, the degree of which can be engineered through controllable
parameters. Our results reproduce the main features of the impurity dynamics
observed by Catani et al. despite experimental thermal effects, and are
supported by simulations of the system in the Gross-Pitaevskii regime.
Moreover, we predict novel effects at lower temperatures due to self-trapping
and the inhomogeneity of the trapped Bose gas.Comment: 7 pages, 3 figure
Brane World in a Topological Black Hole Bulk
We consider a static brane in the background of a topological black hole, in
arbitrary dimensions. For hyperbolic horizons, we find a solution only when the
black hole mass assumes its minimum negative value. In this case, the tension
of the brane vanishes, and the brane position coincides with the location of
the horizon. For an elliptic horizon, we show that the massless mode of
Randall-Sundrum is recovered in the limit of large black hole mass.Comment: Latex, 8 pages, v2: Additional references, to appear in MPL
Bounce and cyclic cosmology in extended nonlinear massive gravity
We investigate non-singular bounce and cyclic cosmological evolutions in a
universe governed by the extended nonlinear massive gravity, in which the
graviton mass is promoted to a scalar-field potential. The extra freedom of the
theory can lead to certain energy conditions violations and drive cyclicity
with two different mechanisms: either with a suitably chosen scalar-field
potential under a given Stuckelberg-scalar function, or with a suitably chosen
Stuckelberg-scalar function under a given scalar-field potential. Our analysis
shows that extended nonlinear massive gravity can alter significantly the
evolution of the universe at both early and late times.Comment: 20 pages, 5 figures, version published at JCA
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