  In the high dimensional regression analysis when the number of predictors is
much larger than the sample size, an important question is to select the
important variable which are relevant to the response variable of interest.
Variable selection and the multiple testing are both tools to address this
issue. However, there is little discussion on the connection of these two
areas. When the signal strength is strong enough such that the selection
consistency is achievable, it seems to be unnecessary to control the false
discovery rate. In this paper, we consider the regime where the signals are
both rare and weak such that the selection consistency is not achievable and
propose a method which controls the false discovery rate asymptotically. It is
theoretically shown that the false non-discovery rate of the proposed method
converges to zero at the optimal rate. Numerical results are provided to
demonstrate the advantage of the proposed method.
