  Hard thresholding, LASSO , adaptive LASSO and SCAD point estimators have been
suggested for use in the linear regression context when most of the components
of the regression parameter vector are believed to be zero, a sparsity type of
assumption. Potscher and Schneider, 2010, Electronic Journal of Statistics,
have considered the properties of fixed-width confidence intervals that include
one of these point estimators (for all possible data values). They consider a
normal linear regression model with orthogonal regressors and show that these
confidence intervals are longer than the standard confidence interval (based on
the maximum likelihood estimator) when the tuning parameter for these point
estimators is chosen to lead to either conservative or consistent model
selection. We extend this analysis to the case of variable-width confidence
intervals that include one of these point estimators (for all possible data
values). In consonance with these findings of Potscher and Schneider, we find
that these confidence intervals perform poorly by comparison with the standard
confidence interval, when the tuning parameter for these point estimators is
chosen to lead to consistent model selection. However, when the tuning
parameter for these point estimators is chosen to lead to conservative model
selection, our conclusions differ from those of Potscher and Schneider. We
consider the variable-width confidence intervals of Farchione and Kabaila,
2008, Statistics & Probability Letters, which have advantages over the standard
confidence interval in the context that there is a belief in a sparsity type of
assumption. These variable-width confidence intervals are shown to include the
hard thresholding, LASSO, adaptive LASSO and SCAD estimators (for all possible
data values) provided that the tuning parameters for these estimators are
chosen to belong to an appropriate interval.
