  Variance estimation in the linear model when $p > n$ is a difficult problem.
Standard least squares estimation techniques do not apply. Several variance
estimators have been proposed in the literature, all with accompanying
asymptotic results proving consistency and asymptotic normality under a variety
of assumptions.
  It is found, however, that most of these estimators suffer large biases in
finite samples when true underlying signals become less sparse with larger per
element signal strength. One estimator seems to be largely neglected in the
literature: a residual sum of squares based estimator using Lasso coefficients
with regularisation parameter selected adaptively (via cross-validation).
  In this paper, we review several variance estimators and perform a reasonably
extensive simulation study in an attempt to compare their finite sample
performance. It would seem from the results that variance estimators with
adaptively chosen regularisation parameters perform admirably over a broad
range of sparsity and signal strength settings. Finally, some intial
theoretical analyses pertaining to these types of estimators are proposed and
developed.
