  Two major research tasks lie at the heart of high dimensional data analysis:
accurate parameter estimation and correct support recovery. The existing
literature mostly aims for either the best parameter estimation or the best
model selection result, however little has been done to understand the
potential interaction between the estimation precision and the selection
behavior. In this work, our minimax result shows that an estimator's
performance of type I error control directly links with its $L_2$ estimation
error rate, and reveals a trade-off phenomenon between the rate of convergence
and the false discovery control: to achieve better accuracy, one risks yielding
more false discoveries. In particular, we characterize the false discovery
control behavior of rate optimal and rate suboptimal estimators under different
sparsity regimes, and discover a rigid dichotomy between these two estimators
under near-linear and linear sparsity settings. In addition, this work provides
a rigorous explanation to the incompatibility phenomenon between selection
consistency and rate minimaxity which has been frequently observed in the high
dimensional literature.
