  Confidence intervals based on penalized maximum likelihood estimators such as
the LASSO, adaptive LASSO, and hard-thresholding are analyzed. In the
known-variance case, the finite-sample coverage properties of such intervals
are determined and it is shown that symmetric intervals are the shortest. The
length of the shortest intervals based on the hard-thresholding estimator is
larger than the length of the shortest interval based on the adaptive LASSO,
which is larger than the length of the shortest interval based on the LASSO,
which in turn is larger than the standard interval based on the maximum
likelihood estimator. In the case where the penalized estimators are tuned to
possess the `sparsity property', the intervals based on these estimators are
larger than the standard interval by an order of magnitude. Furthermore, a
simple asymptotic confidence interval construction in the `sparse' case, that
also applies to the smoothly clipped absolute deviation estimator, is
discussed. The results for the known-variance case are shown to carry over to
the unknown-variance case in an appropriate asymptotic sense.
