697 research outputs found
On posterior distribution of Bayesian wavelet thresholding
We investigate the posterior rate of convergence for wavelet shrinkage using
a Bayesian approach in general Besov spaces. Instead of studying the Bayesian
estimator related to a particular loss function, we focus on the posterior
distribution itself from a nonparametric Bayesian asymptotics point of view and
study its rate of convergence. We obtain the same rate as in
\citet{abramovich04} where the authors studied the convergence of several
Bayesian estimators
Minimax Prediction for Functional Linear Regression with Functional Responses in Reproducing Kernel Hilbert Spaces
In this article, we consider convergence rates in functional linear
regression with functional responses, where the linear coefficient lies in a
reproducing kernel Hilbert space (RKHS). Without assuming that the reproducing
kernel and the covariate covariance kernel are aligned, or assuming polynomial
rate of decay of the eigenvalues of the covariance kernel, convergence rates in
prediction risk are established. The corresponding lower bound in rates is
derived by reducing to the scalar response case. Simulation studies and two
benchmark datasets are used to illustrate that the proposed approach can
significantly outperform the functional PCA approach in prediction
On rates of convergence for posterior distributions under misspecification
We extend the approach of Walker (2003, 2004) to the case of misspecified
models. A sufficient condition for establishing rates of convergence is given
based on a key identity involving martingales, which does not require
construction of tests. We also show roughly that the result obtained by using
tests can also be obtained by our approach, which demonstrates the potential
wider applicability of this method.Comment: 8 pages, no figure
Flexible Shrinkage Estimation in High-Dimensional Varying Coefficient Models
We consider the problem of simultaneous variable selection and constant
coefficient identification in high-dimensional varying coefficient models based
on B-spline basis expansion. Both objectives can be considered as some type of
model selection problems and we show that they can be achieved by a double
shrinkage strategy. We apply the adaptive group Lasso penalty in models
involving a diverging number of covariates, which can be much larger than the
sample size, but we assume the number of relevant variables is smaller than the
sample size via model sparsity. Such so-called ultra-high dimensional settings
are especially challenging in semiparametric models as we consider here and has
not been dealt with before. Under suitable conditions, we show that consistency
in terms of both variable selection and constant coefficient identification can
be achieved, as well as the oracle property of the constant coefficients. Even
in the case that the zero and constant coefficients are known a priori, our
results appear to be new in that it reduces to semivarying coefficient models
(a.k.a. partially linear varying coefficient models) with a diverging number of
covariates. We also theoretically demonstrate the consistency of a
semiparametric BIC-type criterion in this high-dimensional context, extending
several previous results. The finite sample behavior of the estimator is
evaluated by some Monte Carlo studies.Comment: 26 page
Bayesian Nonlinear Principal Component Analysis Using Random Fields
We propose a novel model for nonlinear dimension reduction motivated by the
probabilistic formulation of principal component analysis. Nonlinearity is
achieved by specifying different transformation matrices at different locations
of the latent space and smoothing the transformation using a Markov random
field type prior. The computation is made feasible by the recent advances in
sampling from von Mises-Fisher distributions
Cross Validation for Comparing Multiple Density Estimation Procedures
We demonstrate the consistency of cross validation for comparing multiple
density estimators using simple inequalities on the likelihood ratio. In
nonparametric problems, the splitting of data does not require the domination
of test data over the training/estimation data, contrary to Shao (1993). The
result is complementary to that of Yang (2005) and Yang (2006)
Posterior Convergence and Model Estimation in Bayesian Change-point Problems
We study the posterior distribution of the Bayesian multiple change-point
regression problem when the number and the locations of the change-points are
unknown. While it is relatively easy to apply the general theory to obtain the
rate up to some logarithmic factor, showing the exact
parametric rate of convergence of the posterior distribution requires
additional work and assumptions. Additionally, we demonstrate the asymptotic
normality of the segment levels under these assumptions. For inferences on the
number of change-points, we show that the Bayesian approach can produce a
consistent posterior estimate. Finally, we argue that the point-wise posterior
convergence property as demonstrated might have bad finite sample performance
in that consistent posterior for model selection necessarily implies the
maximal squared risk will be asymptotically larger than the optimal
rate. This is the Bayesian version of the same phenomenon that
has been noted and studied by other authors
Shrinkage Tuning Parameter Selection in Precision Matrices Estimation
Recent literature provides many computational and modeling approaches for
covariance matrices estimation in a penalized Gaussian graphical models but
relatively little study has been carried out on the choice of the tuning
parameter. This paper tries to fill this gap by focusing on the problem of
shrinkage parameter selection when estimating sparse precision matrices using
the penalized likelihood approach. Previous approaches typically used K-fold
cross-validation in this regard. In this paper, we first derived the
generalized approximate cross-validation for tuning parameter selection which
is not only a more computationally efficient alternative, but also achieves
smaller error rate for model fitting compared to leave-one-out
cross-validation. For consistency in the selection of nonzero entries in the
precision matrix, we employ a Bayesian information criterion which provably can
identify the nonzero conditional correlations in the Gaussian model. Our
simulations demonstrate the general superiority of the two proposed selectors
in comparison with leave-one-out cross-validation, ten-fold cross-validation
and Akaike information criterion
Empirical Likelihood Confidence Intervals for Nonparametric Functional Data Analysis
We consider the problem of constructing confidence intervals for
nonparametric functional data analysis using empirical likelihood. In this
doubly infinite-dimensional context, we demonstrate the Wilks's phenomenon and
propose a bias-corrected construction that requires neither undersmoothing nor
direct bias estimation. We also extend our results to partially linear
regression involving functional data. Our numerical results demonstrated the
improved performance of empirical likelihood over approximation based on
asymptotic normality
A simple and efficient algorithm for fused lasso signal approximator with convex loss function
We consider the augmented Lagrangian method (ALM) as a solver for the fused
lasso signal approximator (FLSA) problem. The ALM is a dual method in which
squares of the constraint functions are added as penalties to the Lagrangian.
In order to apply this method to FLSA, two types of auxiliary variables are
introduced to transform the original unconstrained minimization problem into a
linearly constrained minimization problem. Each updating in this iterative
algorithm consists of just a simple one-dimensional convex programming problem,
with closed form solution in many cases. While the existing literature mostly
focused on the quadratic loss function, our algorithm can be easily implemented
for general convex loss. The most attractive feature of this algorithm is its
simplicity in implementation compared to other existing fast solvers. We also
provide some convergence analysis of the algorithm. Finally, the method is
illustrated with some simulation datasets
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