Distributional results for thresholding estimators in high-dimensional
Gaussian regression
abstract:
We study the distribution of hard-, soft-, and adaptive
soft-thresholding estimators within a linear regression model where
the number of parameters k can depend on sample size n and may diverge
with n. In addition to the case of known error-variance, we define and
study versions of the estimators when the error-variance is
unknown. We derive the finite-sample distribution of each estimator
and study its behavior in the large-sample limit, also investigating
the effects of having to estimate the variance when the degrees of
freedom n-k does not tend to infinity or tends to infinity very
slowly. Our analysis encompasses both the case where the estimators
are tuned to perform consistent model selection and the case where the
estimators are tuned to perform conservative model
selection. Furthermore, we discuss consistency, uniform consistency
and derive the uniform convergence rate under either type of tuning.
This is joint work with Benedikt Pötscher.