7,633 research outputs found
Aggregation of Nonparametric Estimators for Volatility Matrix
An aggregated method of nonparametric estimators based on time-domain and
state-domain estimators is proposed and studied. To attenuate the curse of
dimensionality, we propose a factor modeling strategy. We first investigate the
asymptotic behavior of nonparametric estimators of the volatility matrix in the
time domain and in the state domain. Asymptotic normality is separately
established for nonparametric estimators in the time domain and state domain.
These two estimators are asymptotically independent. Hence, they can be
combined, through a dynamic weighting scheme, to improve the efficiency of
volatility matrix estimation. The optimal dynamic weights are derived, and it
is shown that the aggregated estimator uniformly dominates volatility matrix
estimators using time-domain or state-domain smoothing alone. A simulation
study, based on an essentially affine model for the term structure, is
conducted, and it demonstrates convincingly that the newly proposed procedure
outperforms both time- and state-domain estimators. Empirical studies further
endorse the advantages of our aggregated method.Comment: 46 pages, 11 PostScript figure
Asymptotic properties for combined and concave regularization
Two important goals of high-dimensional modeling are prediction and variable
selection. In this article, we consider regularization with combined and
concave penalties, and study the sampling properties of the global optimum of
the suggested method in ultra-high dimensional settings. The -penalty
provides the minimum regularization needed for removing noise variables in
order to achieve oracle prediction risk, while concave penalty imposes
additional regularization to control model sparsity. In the linear model
setting, we prove that the global optimum of our method enjoys the same oracle
inequalities as the lasso estimator and admits an explicit bound on the false
sign rate, which can be asymptotically vanishing. Moreover, we establish oracle
risk inequalities for the method and the sampling properties of computable
solutions. Numerical studies suggest that our method yields more stable
estimates than using a concave penalty alone.Comment: 16 page
Asymptotic equivalence of regularization methods in thresholded parameter space
High-dimensional data analysis has motivated a spectrum of regularization
methods for variable selection and sparse modeling, with two popular classes of
convex ones and concave ones. A long debate has been on whether one class
dominates the other, an important question both in theory and to practitioners.
In this paper, we characterize the asymptotic equivalence of regularization
methods, with general penalty functions, in a thresholded parameter space under
the generalized linear model setting, where the dimensionality can grow up to
exponentially with the sample size. To assess their performance, we establish
the oracle inequalities, as in Bickel, Ritov and Tsybakov (2009), of the global
minimizer for these methods under various prediction and variable selection
losses. These results reveal an interesting phase transition phenomenon. For
polynomially growing dimensionality, the -regularization method of Lasso
and concave methods are asymptotically equivalent, having the same convergence
rates in the oracle inequalities. For exponentially growing dimensionality,
concave methods are asymptotically equivalent but have faster convergence rates
than the Lasso. We also establish a stronger property of the oracle risk
inequalities of the regularization methods, as well as the sampling properties
of computable solutions. Our new theoretical results are illustrated and
justified by simulation and real data examples.Comment: 39 pages, 3 figure
Non-Concave Penalized Likelihood with NP-Dimensionality
Penalized likelihood methods are fundamental to ultra-high dimensional
variable selection. How high dimensionality such methods can handle remains
largely unknown. In this paper, we show that in the context of generalized
linear models, such methods possess model selection consistency with oracle
properties even for dimensionality of Non-Polynomial (NP) order of sample size,
for a class of penalized likelihood approaches using folded-concave penalty
functions, which were introduced to ameliorate the bias problems of convex
penalty functions. This fills a long-standing gap in the literature where the
dimensionality is allowed to grow slowly with the sample size. Our results are
also applicable to penalized likelihood with the -penalty, which is a
convex function at the boundary of the class of folded-concave penalty
functions under consideration. The coordinate optimization is implemented for
finding the solution paths, whose performance is evaluated by a few simulation
examples and the real data analysis.Comment: 37 pages, 2 figure
Innovated scalable efficient estimation in ultra-large Gaussian graphical models
Large-scale precision matrix estimation is of fundamental importance yet
challenging in many contemporary applications for recovering Gaussian graphical
models. In this paper, we suggest a new approach of innovated scalable
efficient estimation (ISEE) for estimating large precision matrix. Motivated by
the innovated transformation, we convert the original problem into that of
large covariance matrix estimation. The suggested method combines the strengths
of recent advances in high-dimensional sparse modeling and large covariance
matrix estimation. Compared to existing approaches, our method is scalable and
can deal with much larger precision matrices with simple tuning. Under mild
regularity conditions, we establish that this procedure can recover the
underlying graphical structure with significant probability and provide
efficient estimation of link strengths. Both computational and theoretical
advantages of the procedure are evidenced through simulation and real data
examples.Comment: to appear, The Annals of Statistics (2016
Sure Independence Screening for Ultra-High Dimensional Feature Space
Variable selection plays an important role in high dimensional statistical
modeling which nowadays appears in many areas and is key to various scientific
discoveries. For problems of large scale or dimensionality , estimation
accuracy and computational cost are two top concerns. In a recent paper, Candes
and Tao (2007) propose the Dantzig selector using regularization and show
that it achieves the ideal risk up to a logarithmic factor . Their
innovative procedure and remarkable result are challenged when the
dimensionality is ultra high as the factor can be large and their
uniform uncertainty principle can fail.
Motivated by these concerns, we introduce the concept of sure screening and
propose a sure screening method based on a correlation learning, called the
Sure Independence Screening (SIS), to reduce dimensionality from high to a
moderate scale that is below sample size. In a fairly general asymptotic
framework, the correlation learning is shown to have the sure screening
property for even exponentially growing dimensionality. As a methodological
extension, an iterative SIS (ISIS) is also proposed to enhance its finite
sample performance. With dimension reduced accurately from high to below sample
size, variable selection can be improved on both speed and accuracy, and can
then be accomplished by a well-developed method such as the SCAD, Dantzig
selector, Lasso, or adaptive Lasso. The connections of these penalized
least-squares methods are also elucidated.Comment: 43 pages, 6 PostScript figure
Asymptotic Theory of Eigenvectors for Large Random Matrices
Characterizing the exact asymptotic distributions of high-dimensional
eigenvectors for large structured random matrices poses important challenges
yet can provide useful insights into a range of applications. To this end, in
this paper we introduce a general framework of asymptotic theory of
eigenvectors (ATE) for large structured symmetric random matrices with
heterogeneous variances, and establish the asymptotic properties of the spiked
eigenvectors and eigenvalues for the scenario of the generalized Wigner matrix
noise, where the mean matrix is assumed to have the low-rank structure. Under
some mild regularity conditions, we provide the asymptotic expansions for the
spiked eigenvalues and show that they are asymptotically normal after some
normalization. For the spiked eigenvectors, we establish novel asymptotic
expansions for the general linear combination and further show that it is
asymptotically normal after some normalization, where the weight vector can be
arbitrary. We also provide a more general asymptotic theory for the spiked
eigenvectors using the bilinear form. Simulation studies verify the validity of
our new theoretical results. Our family of models encompasses many popularly
used ones such as the stochastic block models with or without overlapping
communities for network analysis and the topic models for text analysis, and
our general theory can be exploited for statistical inference in these
large-scale applications.Comment: 67 pages, 3 figure
High dimensional thresholded regression and shrinkage effect
High-dimensional sparse modeling via regularization provides a powerful tool
for analyzing large-scale data sets and obtaining meaningful, interpretable
models. The use of nonconvex penalty functions shows advantage in selecting
important features in high dimensions, but the global optimality of such
methods still demands more understanding. In this paper, we consider sparse
regression with hard-thresholding penalty, which we show to give rise to
thresholded regression. This approach is motivated by its close connection with
the -regularization, which can be unrealistic to implement in practice but
of appealing sampling properties, and its computational advantage. Under some
mild regularity conditions allowing possibly exponentially growing
dimensionality, we establish the oracle inequalities of the resulting
regularized estimator, as the global minimizer, under various prediction and
variable selection losses, as well as the oracle risk inequalities of the
hard-thresholded estimator followed by a further -regularization. The risk
properties exhibit interesting shrinkage effects under both estimation and
prediction losses. We identify the optimal choice of the ridge parameter, which
is shown to have simultaneous advantages to both the -loss and prediction
loss. These new results and phenomena are evidenced by simulation and real data
examples.Comment: 23 pages, 3 figures, 5 table
Dynamic Principles of Center of Mass in Human Walking
We present results of an analytic and numerical calculation that studies the
relationship between the time of initial foot contact and the ground reaction
force of human gait and explores the dynamic principle of center of mass.
Assuming the ground reaction force of both feet to be the same in the same
phase of a stride cycle, we establish the relationships between the time of
initial foot contact and the ground reaction force, acceleration, velocity,
displacement and average kinetic energy of center of mass. We employ the
dispersion to analyze the effect of the time of the initial foot contact that
imposes upon these physical quantities. Our study reveals that when the time of
one foot's initial contact falls right in the middle of the other foot's stride
cycle, these physical quantities reach extrema. An action function has been
identified as the dispersion of the physical quantities and optimized analysis
used to prove the least-action principle in gait. In addition to being very
significant to the research domains such as clinical diagnosis, biped robot's
gait control, the exploration of this principle can simplify our understanding
of the basic properties of gait.Comment: 16 pages, 5 figure
Nonuniformity of P-values Can Occur Early in Diverging Dimensions
Evaluating the joint significance of covariates is of fundamental importance
in a wide range of applications. To this end, p-values are frequently employed
and produced by algorithms that are powered by classical large-sample
asymptotic theory. It is well known that the conventional p-values in Gaussian
linear model are valid even when the dimensionality is a non-vanishing fraction
of the sample size, but can break down when the design matrix becomes singular
in higher dimensions or when the error distribution deviates from Gaussianity.
A natural question is when the conventional p-values in generalized linear
models become invalid in diverging dimensions. We establish that such a
breakdown can occur early in nonlinear models. Our theoretical
characterizations are confirmed by simulation studies.Comment: 23 pages including 8 figure
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