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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
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