47,381 research outputs found
SCAD-penalized regression in high-dimensional partially linear models
We consider the problem of simultaneous variable selection and estimation in
partially linear models with a divergent number of covariates in the linear
part, under the assumption that the vector of regression coefficients is
sparse. We apply the SCAD penalty to achieve sparsity in the linear part and
use polynomial splines to estimate the nonparametric component. Under
reasonable conditions, it is shown that consistency in terms of variable
selection and estimation can be achieved simultaneously for the linear and
nonparametric components. Furthermore, the SCAD-penalized estimators of the
nonzero coefficients are shown to have the asymptotic oracle property, in the
sense that it is asymptotically normal with the same means and covariances that
they would have if the zero coefficients were known in advance. The finite
sample behavior of the SCAD-penalized estimators is evaluated with simulation
and illustrated with a data set.Comment: Published in at http://dx.doi.org/10.1214/07-AOS580 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On Bayesian Oracle Properties
When model uncertainty is handled by Bayesian model averaging (BMA) or
Bayesian model selection (BMS), the posterior distribution possesses a
desirable "oracle property" for parametric inference, if for large enough data
it is nearly as good as the oracle posterior, obtained by assuming
unrealistically that the true model is known and only the true model is used.
We study the oracle properties in a very general context of quasi-posterior,
which can accommodate non-regular models with cubic root asymptotics and
partial identification. Our approach for proving the oracle properties is based
on a unified treatment that bounds the posterior probability of model
mis-selection. This theoretical framework can be of interest to Bayesian
statisticians who would like to theoretically justify their new model selection
or model averaging methods in addition to empirical results. Furthermore, for
non-regular models, we obtain nontrivial conclusions on the choice of prior
penalty on model complexity, the temperature parameter of the quasi-posterior,
and the advantage of BMA over BMS.Comment: 31 page
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
A Selective Review of Group Selection in High-Dimensional Models
Grouping structures arise naturally in many statistical modeling problems.
Several methods have been proposed for variable selection that respect grouping
structure in variables. Examples include the group LASSO and several concave
group selection methods. In this article, we give a selective review of group
selection concerning methodological developments, theoretical properties and
computational algorithms. We pay particular attention to group selection
methods involving concave penalties. We address both group selection and
bi-level selection methods. We describe several applications of these methods
in nonparametric additive models, semiparametric regression, seemingly
unrelated regressions, genomic data analysis and genome wide association
studies. We also highlight some issues that require further study.Comment: Published in at http://dx.doi.org/10.1214/12-STS392 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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