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Statistics Seminar Series
Thursday, Dec. 13, 2012, 4:00 PM
Cullimore, Room 111
New Jersey Institute of Technology
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Residual Variance and the Signal-to-noise Ratio in High-dimensional
Linear Models
Lee Dicker
Department of Statistics, Rutgers University
Abstract
Residual variance and the signal-to-noise ratio are important quantities in many
statistical models and model fitting procedures. They play an important role in
regression diagnostics, in determining
the performance limits in estimation and prediction problems, and in shrinkage
parameter selection in many popular regularized regression methods for
high-dimensional data analysis. We propose new estimators for the residual
variance, the l2-signal strength, and the signal-to-noise ratio that
are consistent and asymptotically normal in high-dimensional linear models with
Gaussian predictors and errors,
where the number of predictors,
d,
is proportional to the number of observations,
n.
Existing results on residual variance estimation in high-dimensional linear
models depend on sparsity in the underlying
signal. Our results require no sparsity assumptions and imply that the residual
variance may be consistently estimated even when
d > n
and the underlying signal itself is non-estimable. Basic numerical work
suggests that some of the distributional assumptions made for our theoretical
results may be relaxed.