  Choosing a shrinkage method can be done by selecting a penalty from a list of
pre-specified penalties or by constructing a penalty based on the data. If a
list of penalties for a class of linear models is given, we provide comparisons
based on sample size and number of non-zero parameters under a predictive
stability criterion based on data perturbation. These comparisons provide
recommendations for penalty selection in a variety of settings. If the
preference is to construct a penalty customized for a given problem, then we
propose a technique based on genetic algorithms, again using a predictive
criterion. We find that, in general, a custom penalty never performs worse than
any commonly used penalties but that there are cases the custom penalty reduces
to a recognizable penalty. Since penalty selection is mathematically equivalent
to prior selection, our method also constructs priors.
  The techniques and recommendations we offer are intended for finite sample
cases. In this context, we argue that predictive stability under perturbation
is one of the few relevant properties that can be invoked when the true model
is not known. Nevertheless, we study variable inclusion in simulations and, as
part of our shrinkage selection strategy, we include oracle property
considerations. In particular, we see that the oracle property typically holds
for penalties that satisfy basic regularity conditions and therefore is not
restrictive enough to play a direct role in penalty selection. In addition, our
real data example also includes considerations merging from model
mis-specification.
