78 research outputs found
Deconvolution Estimation in Measurement Error Models: The R Package decon
Data from many scientific areas often come with measurement error. Density or distribution function estimation from contaminated data and nonparametric regression with errors in variables are two important topics in measurement error models. In this paper, we present a new software package decon for R, which contains a collection of functions that use the deconvolution kernel methods to deal with the measurement error problems. The functions allow the errors to be either homoscedastic or heteroscedastic. To make the deconvolution estimators computationally more efficient in R, we adapt the fast Fourier transform algorithm for density estimation with error-free data to the deconvolution kernel estimation. We discuss the practical selection of the smoothing parameter in deconvolution methods and illustrate the use of the package through both simulated and real examples.
Deconvolution Estimation in Measurement Error Models: The R Package decon
Data from many scientific areas often come with measurement error. Density or distribution function estimation from contaminated data and nonparametric regression with errors in variables are two important topics in measurement error models. In this paper, we present a new software package decon for R, which contains a collection of functions that use the deconvolution kernel methods to deal with the measurement error problems. The functions allow the errors to be either homoscedastic or heteroscedastic. To make the deconvolution estimators computationally more efficient in R, we adapt the fast Fourier transform algorithm for density estimation with error-free data to the deconvolution kernel estimation. We discuss the practical selection of the smoothing parameter in deconvolution methods and illustrate the use of the package through both simulated and real examples
Bandwidth selection in kernel empirical risk minimization via the gradient
In this paper, we deal with the data-driven selection of multidimensional and
possibly anisotropic bandwidths in the general framework of kernel empirical
risk minimization. We propose a universal selection rule, which leads to
optimal adaptive results in a large variety of statistical models such as
nonparametric robust regression and statistical learning with errors in
variables. These results are stated in the context of smooth loss functions,
where the gradient of the risk appears as a good criterion to measure the
performance of our estimators. The selection rule consists of a comparison of
gradient empirical risks. It can be viewed as a nontrivial improvement of the
so-called Goldenshluger-Lepski method to nonlinear estimators. Furthermore, one
main advantage of our selection rule is the nondependency on the Hessian matrix
of the risk, usually involved in standard adaptive procedures.Comment: Published at http://dx.doi.org/10.1214/15-AOS1318 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Laplace deconvolution with noisy observations
In the present paper we consider Laplace deconvolution for discrete noisy
data observed on the interval whose length may increase with a sample size.
Although this problem arises in a variety of applications, to the best of our
knowledge, it has been given very little attention by the statistical
community. Our objective is to fill this gap and provide statistical treatment
of Laplace deconvolution problem with noisy discrete data. The main
contribution of the paper is explicit construction of an asymptotically
rate-optimal (in the minimax sense) Laplace deconvolution estimator which is
adaptive to the regularity of the unknown function. We show that the original
Laplace deconvolution problem can be reduced to nonparametric estimation of a
regression function and its derivatives on the interval of growing length T_n.
Whereas the forms of the estimators remain standard, the choices of the
parameters and the minimax convergence rates, which are expressed in terms of
T_n^2/n in this case, are affected by the asymptotic growth of the length of
the interval.
We derive an adaptive kernel estimator of the function of interest, and
establish its asymptotic minimaxity over a range of Sobolev classes. We
illustrate the theory by examples of construction of explicit expressions of
Laplace deconvolution estimators. A simulation study shows that, in addition to
providing asymptotic optimality as the number of observations turns to
infinity, the proposed estimator demonstrates good performance in finite sample
examples
Variable selection in measurement error models
Measurement error data or errors-in-variable data have been collected in many
studies. Natural criterion functions are often unavailable for general
functional measurement error models due to the lack of information on the
distribution of the unobservable covariates. Typically, the parameter
estimation is via solving estimating equations. In addition, the construction
of such estimating equations routinely requires solving integral equations,
hence the computation is often much more intensive compared with ordinary
regression models. Because of these difficulties, traditional best subset
variable selection procedures are not applicable, and in the measurement error
model context, variable selection remains an unsolved issue. In this paper, we
develop a framework for variable selection in measurement error models via
penalized estimating equations. We first propose a class of selection
procedures for general parametric measurement error models and for general
semi-parametric measurement error models, and study the asymptotic properties
of the proposed procedures. Then, under certain regularity conditions and with
a properly chosen regularization parameter, we demonstrate that the proposed
procedure performs as well as an oracle procedure. We assess the finite sample
performance via Monte Carlo simulation studies and illustrate the proposed
methodology through the empirical analysis of a familiar data set.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ205 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Recent advances in directional statistics
Mainstream statistical methodology is generally applicable to data observed
in Euclidean space. There are, however, numerous contexts of considerable
scientific interest in which the natural supports for the data under
consideration are Riemannian manifolds like the unit circle, torus, sphere and
their extensions. Typically, such data can be represented using one or more
directions, and directional statistics is the branch of statistics that deals
with their analysis. In this paper we provide a review of the many recent
developments in the field since the publication of Mardia and Jupp (1999),
still the most comprehensive text on directional statistics. Many of those
developments have been stimulated by interesting applications in fields as
diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics,
image analysis, text mining, environmetrics, and machine learning. We begin by
considering developments for the exploratory analysis of directional data
before progressing to distributional models, general approaches to inference,
hypothesis testing, regression, nonparametric curve estimation, methods for
dimension reduction, classification and clustering, and the modelling of time
series, spatial and spatio-temporal data. An overview of currently available
software for analysing directional data is also provided, and potential future
developments discussed.Comment: 61 page
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VARIATIONAL APPROXIMATIONS FOR DENSITY DECONVOLUTION
This thesis considers the problem of density estimation when the variables of interest are subject to measurement error. The measurement error is assumed to be additive and homoscedastic. We specify the density of interest by a Dirichlet Process Mixture Model and establish variational approximation approaches to the density deconvolution problem. Gaussian and Laplacian error distributions are considered, which are representatives of supersmooth and ordinary smooth distributions, respectively. We develop two variational approximation algorithms for Gaussian error deconvolution and one variational approximation algorithm for Laplacian error deconvolution. Their performances are compared to deconvoluting kernels and Monte Carlo Markov Chain method by simulation experiments. A conjecture based on hidden variables categorization is proposed to explain why two variational approximation algorithms for Gaussian error deconvolution perform differently. We establish a stochastic variational approximation algorithm for Gaussian error deconvolution, which improves the performance of variational approximation algorithm and performs as well as MCMC method at faster speed. The stochastic variational approximation algorithm is applied to simulation experiments and an example of physical activity measurements
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