4,109 research outputs found
Random action of compact Lie groups and minimax estimation of a mean pattern
This paper considers the problem of estimating a mean pattern in the setting
of Grenander's pattern theory. Shape variability in a data set of curves or
images is modeled by the random action of elements in a compact Lie group on an
infinite dimensional space. In the case of observations contaminated by an
additive Gaussian white noise, it is shown that estimating a reference template
in the setting of Grenander's pattern theory falls into the category of
deconvolution problems over Lie groups. To obtain this result, we build an
estimator of a mean pattern by using Fourier deconvolution and harmonic
analysis on compact Lie groups. In an asymptotic setting where the number of
observed curves or images tends to infinity, we derive upper and lower bounds
for the minimax quadratic risk over Sobolev balls. This rate depends on the
smoothness of the density of the random Lie group elements representing shape
variability in the data, which makes a connection between estimating a mean
pattern and standard deconvolution problems in nonparametric statistics
Intensity estimation of non-homogeneous Poisson processes from shifted trajectories
This paper considers the problem of adaptive estimation of a non-homogeneous
intensity function from the observation of n independent Poisson processes
having a common intensity that is randomly shifted for each observed
trajectory. 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
Data-driven efficient score tests for deconvolution problems
We consider testing statistical hypotheses about densities of signals in
deconvolution models. A new approach to this problem is proposed. We
constructed score tests for the deconvolution with the known noise density and
efficient score tests for the case of unknown density. The tests are
incorporated with model selection rules to choose reasonable model dimensions
automatically by the data. Consistency of the tests is proved
Methodology and convergence rates for functional linear regression
In functional linear regression, the slope ``parameter'' is a function.
Therefore, in a nonparametric context, it is determined by an infinite number
of unknowns. Its estimation involves solving an ill-posed problem and has
points of contact with a range of methodologies, including statistical
smoothing and deconvolution. The standard approach to estimating the slope
function is based explicitly on functional principal components analysis and,
consequently, on spectral decomposition in terms of eigenvalues and
eigenfunctions. We discuss this approach in detail and show that in certain
circumstances, optimal convergence rates are achieved by the PCA technique. An
alternative approach based on quadratic regularisation is suggested and shown
to have advantages from some points of view.Comment: Published at http://dx.doi.org/10.1214/009053606000000957 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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