  Given a regression dataset of size $n$, most of the classical model selection
literature treats the problem of selecting a subset of covariates to be used
for prediction of future responses. In this paper we assume that a sample of
$\cal J$ regression datasets of different sizes from a population of datasets
having the same set of covariates and responses is observed. The goal is to
select, for each $n$, a single subset of covariates to be used for prediction
of future responses for any dataset of size $n$ from the population (which may
or may not be in the sample of datasets). The regression coefficients used in
the prediction are estimated using the $n$ observations consisting of
covariates and responses in the sample for which prediction of future responses
is to be done, and thus they differ across different samples. For example, if
the response is a diagnosis, and the covariates are medical background
variables and measurements, the goal is to select a standard set of
measurements for different clinics, say, where each clinic may estimate and use
its own coefficients for prediction (depending on local conditions, prevalence,
etc.). The selected subset naturally depends on the sample size $n$, with a
larger sample size allowing a more elaborate model. Since we consider
prediction for any (or a randomly chosen) dataset in the population, it is
natural to consider random covariates. If the population consists of datasets
that are similar, our approach amounts to borrowing information, leading to a
subset selection that is efficient for prediction. On the other hand, if the
datasets are dissimilar, then our goal is to find a "compromise" subset of
covariates for the different regressions.
