Subsampling estimates of the Lasso distribution.

Abstract
We investigate possibilities offered by subsampling to etimate the distribution of the Lasso
estimator and construct confidence intervals/hypothesis tests. Despite being inferior to
the bootstrap in terms of higher-order accuracy in situations where the later is consistent,
subsampling offers the advantage to work under very weak assumptions. Thus, building
upon Knight and Fu (2000), we first study the asymptotics of the Lasso estimator in a
low dimensional setting and prove that under an orthogonal design assumption, the finite
sample component distributions converge to a limit in a mode allowing for consistency of
subsampling confidence intervals. We give hints that this result holds in greater generality.
In a high dimensional setting, we study the adaptive Lasso under assumption of partial
orthogonality introduced by Huang, Ma, and Zhang (2008) and use the partial oracle result
in distribution to argue that subsampling should provide valid confidence intervals for
nonzero parameters. Simulations studies confirm the validity of subsampling to construct
confidence intervals, tests for null hypotheses ansd control the FWER through subsampled
p-values in a low dimensional setting. In the high dimensional setting, confidence intervals
for nonzero coefficients are slightly anticonservative and false positive rates are shown to
be conservative.
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