  Recent results concerning asymptotic Bayes-optimality under sparsity (ABOS)
of multiple testing procedures are extended to fairly generally distributed
effect sizes under the alternative. An asymptotic framework is considered where
both the number of tests m and the sample size m go to infinity, while the
fraction p of true alternatives converges to zero. It is shown that under mild
restrictions on the loss function nontrivial asymptotic inference is possible
only if n increases to infinity at least at the rate of log m. Based on this
assumption precise conditions are given under which the Bonferroni correction
with nominal Family Wise Error Rate (FWER) level alpha and the Benjamini-
Hochberg procedure (BH) at FDR level alpha are asymptotically optimal. When n
is proportional to log m then alpha can remain fixed, whereas when n increases
to infinity at a quicker rate, then alpha has to converge to zero roughly like
n^(-1/2). Under these conditions the Bonferroni correction is ABOS in case of
extreme sparsity, while BH adapts well to the unknown level of sparsity.
  In the second part of this article these optimality results are carried over
to model selection in the context of multiple regression with orthogonal
regressors. Several modifications of Bayesian Information Criterion are
considered, controlling either FWER or FDR, and conditions are provided under
which these selection criteria are ABOS. Finally the performance of these
criteria is examined in a brief simulation study.
