4,384 research outputs found
A deconvolution approach to estimation of a common shape in a shifted curves model
This paper considers the problem of adaptive estimation of a mean pattern in a randomly shifted curve model. We show that this problem can be transformed into a linear inverse problem, where the density of the random shifts plays the role of a convolution operator. An adaptive estimator of the mean pattern, based on wavelet thresholding is proposed. We study its consistency for the quadratic risk as the number of observed curves tends to infinity, and this estimator is shown to achieve a near-minimax rate of convergence over a large class of Besov balls. This rate depends both on the smoothness of the common shape of the curves and on the decay of the Fourier coefficients of the density of the random shifts. Hence, this paper makes a connection between mean pattern estimation and the statistical analysis of linear inverse problems, which is a new point of view on curve registration and image warping problems. We also provide a new method to estimate the unknown random shifts between curves. Some numerical experiments are given to illustrate the performances of our approach and to compare them with another algorithm existing in the literature
Block thresholding for wavelet-based estimation of function derivatives from a heteroscedastic multichannel convolution model
We observe heteroscedastic stochastic processes , where
for any and , is the convolution
product of an unknown function and a known blurring function
corrupted by Gaussian noise. Under an ordinary smoothness assumption on
, our goal is to estimate the -th derivatives (in weak
sense) of from the observations. We propose an adaptive estimator based on
wavelet block thresholding, namely the "BlockJS estimator". Taking the mean
integrated squared error (MISE), our main theoretical result investigates the
minimax rates over Besov smoothness spaces, and shows that our block estimator
can achieve the optimal minimax rate, or is at least nearly-minimax in the
least favorable situation. We also report a comprehensive suite of numerical
simulations to support our theoretical findings. The practical performance of
our block estimator compares very favorably to existing methods of the
literature on a large set of test functions
Nonparametric methods for volatility density estimation
Stochastic volatility modelling of financial processes has become
increasingly popular. The proposed models usually contain a stationary
volatility process. We will motivate and review several nonparametric methods
for estimation of the density of the volatility process. Both models based on
discretely sampled continuous time processes and discrete time models will be
discussed.
The key insight for the analysis is a transformation of the volatility
density estimation problem to a deconvolution model for which standard methods
exist. Three type of nonparametric density estimators are reviewed: the
Fourier-type deconvolution kernel density estimator, a wavelet deconvolution
density estimator and a penalized projection estimator. The performance of
these estimators will be compared. Key words: stochastic volatility models,
deconvolution, density estimation, kernel estimator, wavelets, minimum contrast
estimation, mixin
Log-Density Deconvolution by Wavelet Thresholding
This paper proposes a new wavelet-based method for deconvolving a density. The estimator combines the ideas of nonlinear wavelet thresholding with periodised Meyer wavelets and estimation by information projection. It is guaranteed to be in the class of density functions, in particular it is positive everywhere by construction. The asymptotic optimality of the estimator is established in terms of rate of convergence of the Kullback-Leibler discrepancy over Besov classes. Finite sample properties is investigated in detail, and show the excellent empirical performance of the estimator, compared with other recently introduced estimators.deconvolution, wavelet thresholding,adaptive estimation
Intensity estimation of non-homogeneous Poisson processes from shifted trajectories
In this paper, we consider the problem of estimating nonparametrically a mean pattern intensity λ from the observation of n independent and non-homogeneous Poisson processes N1,…,Nn on the interval [0,1]. This problem arises when data (counts) are collected independently from n individuals according to similar Poisson processes. We show that estimating this intensity is a deconvolution problem for which the density of the random shifts plays the role of the convolution operator. In an asymptotic setting where the number n of observed trajectories tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Besov balls. Non-linear thresholding in a Meyer wavelet basis is used to derive an adaptive estimator of the intensity. The proposed estimator is shown to achieve a near-minimax rate of convergence. This rate depends both on the smoothness of the intensity function and the density of the random shifts, which makes a connection between the classical deconvolution problem in nonparametric statistics and the estimation of a mean intensity from the observations of independent Poisson processes
Estimation of Primordial Spectrum with post-WMAP 3 year data
In this paper we implement an improved (error sensitive) Richardson-Lucy
deconvolution algorithm on the measured angular power spectrum from the WMAP 3
year data to determine the primordial power spectrum assuming different points
in the cosmological parameter space for a flat LCDM cosmological model. We also
present the preliminary results of the cosmological parameter estimation by
assuming a free form of the primordial spectrum, for a reasonably large volume
of the parameter space. The recovered spectrum for a considerably large number
of the points in the cosmological parameter space has a likelihood far better
than a `best fit' power law spectrum up to \Delta \chi^2_{eff} \approx -30. We
use Discrete Wavelet Transform (DWT) for smoothing the raw recovered spectrum
from the binned data. The results obtained here reconfirm and sharpen the
conclusion drawn from our previous analysis of the WMAP 1st year data. A sharp
cut off around the horizon scale and a bump after the horizon scale seem to be
a common feature for all of these reconstructed primordial spectra. We have
shown that although the WMAP 3 year data prefers a lower value of matter
density for a power law form of the primordial spectrum, for a free form of the
spectrum, we can get a very good likelihood to the data for higher values of
matter density. We have also shown that even a flat CDM model, allowing a free
form of the primordial spectrum, can give a very high likelihood fit to the
data. Theoretical interpretation of the results is open to the cosmology
community. However, this work provides strong evidence that the data retains
discriminatory power in the cosmological parameter space even when there is
full freedom in choosing the primordial spectrum.Comment: 13 pages, 4 figures, uses Revtex4, new analysis and results,
references added, matches version accepted to Phys. Rev.
Functional deconvolution in a periodic setting: Uniform case
We extend deconvolution in a periodic setting to deal with functional data.
The resulting functional deconvolution model can be viewed as a generalization
of a multitude of inverse problems in mathematical physics where one needs to
recover initial or boundary conditions on the basis of observations from a
noisy solution of a partial differential equation. In the case when it is
observed at a finite number of distinct points, the proposed functional
deconvolution model can also be viewed as a multichannel deconvolution model.
We derive minimax lower bounds for the -risk in the proposed functional
deconvolution model when is assumed to belong to a Besov ball and
the blurring function is assumed to possess some smoothness properties,
including both regular-smooth and super-smooth convolutions. Furthermore, we
propose an adaptive wavelet estimator of that is asymptotically
optimal (in the minimax sense), or near-optimal within a logarithmic factor, in
a wide range of Besov balls. In addition, we consider a discretization of the
proposed functional deconvolution model and investigate when the availability
of continuous data gives advantages over observations at the asymptotically
large number of points. As an illustration, we discuss particular examples for
both continuous and discrete settings.Comment: Published in at http://dx.doi.org/10.1214/07-AOS552 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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