  When dealing with the problem of simultaneously testing a large number of
null hypotheses, a natural testing strategy is to first reduce the number of
tested hypotheses by some selection (screening or filtering) process, and then
to simultaneously test the selected hypotheses. The main advantage of this
strategy is to greatly reduce the severe effect of high dimensions. However,
the first screening or selection stage must be properly accounted for in order
to maintain some type of error control. In this paper, we will introduce a
selection rule based on a selection statistic that is independent of the test
statistic when the tested hypothesis is true. Combining this selection rule and
the conventional Bonferroni procedure, we can develop a powerful and valid
two-stage procedure. The introduced procedure has several nice properties: (i)
it completely removes the selection effect; (ii) it reduces the multiplicity
effect; (iii) it does not "waste" data while carrying out both selection and
testing. Asymptotic power analysis and simulation studies illustrate that this
proposed method can provide higher power compared to usual multiple testing
methods while controlling the Type 1 error rate. Optimal selection thresholds
are also derived based on our asymptotic analysis.
