199 research outputs found
Estimation in high-dimensional linear models with deterministic design matrices
Because of the advance in technologies, modern statistical studies often
encounter linear models with the number of explanatory variables much larger
than the sample size. Estimation and variable selection in these
high-dimensional problems with deterministic design points is very different
from those in the case of random covariates, due to the identifiability of the
high-dimensional regression parameter vector. We show that a reasonable
approach is to focus on the projection of the regression parameter vector onto
the linear space generated by the design matrix. In this work, we consider the
ridge regression estimator of the projection vector and propose to threshold
the ridge regression estimator when the projection vector is sparse in the
sense that many of its components are small. The proposed estimator has an
explicit form and is easy to use in application. Asymptotic properties such as
the consistency of variable selection and estimation and the convergence rate
of the prediction mean squared error are established under some sparsity
conditions on the projection vector. A simulation study is also conducted to
examine the performance of the proposed estimator.Comment: Published in at http://dx.doi.org/10.1214/12-AOS982 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Level Set Dynamics and the Non-blowup of the 2D Quasi-geostrophic Equation
In this article we apply the technique proposed in Deng-Hou-Yu (Comm. PDE,
2005) to study the level set dynamics of the 2D quasi-geostrophic equation.
Under certain assumptions on the local geometric regularity of the level sets
of , we obtain global regularity results with improved growth estimate
on . We further perform numerical simulations to
study the local geometric properties of the level sets near the region of
maximum . The numerical results indicate that the
assumptions on the local geometric regularity of the level sets of in
our theorems are satisfied. Therefore these theorems provide a good explanation
of the double exponential growth of observed in this
and past numerical simulations.Comment: 25 pages, 10 figures. Corrected a few typo
On Variable Ordination of Modified Cholesky Decomposition for Sparse Covariance Matrix Estimation
Estimation of large sparse covariance matrices is of great importance for
statistical analysis, especially in the high-dimensional settings. The
traditional approach such as the sample covariance matrix performs poorly due
to the high dimensionality. The modified Cholesky decomposition (MCD) is a
commonly used method for sparse covariance matrix estimation. However, the MCD
method relies on the order of variables, which is often not available or cannot
be pre-determined in practice. In this work, we solve this order issue by
obtaining a set of covariance matrix estimates under different orders of
variables used in the MCD. Then we consider an ensemble estimator as the
"center" of such a set of covariance matrix estimates with respect to the
Frobenius norm. The proposed method not only ensures the estimator to be
positive definite, but also can capture the underlying sparse structure of the
covariance matrix. Under some weak regularity conditions, we establish both
algorithmic convergence and asymptotical convergence of the proposed method.
The merits of the proposed method are illustrated through simulation studies
and one real data example
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