  The generalized linear models (GLM) have been widely used in practice to
model non-Gaussian response variables. When the number of explanatory features
is relatively large, scientific researchers are of interest to perform
controlled feature selection in order to simplify the downstream analysis. This
paper introduces a new framework for feature selection in GLMs that can achieve
false discovery rate (FDR) control in two asymptotic regimes. The key step is
to construct a mirror statistic to measure the importance of each feature,
which is based upon two (asymptotically) independent estimates of the
corresponding true coefficient obtained via either the data-splitting method or
the Gaussian mirror method. The FDR control is achieved by taking advantage of
the mirror statistic's property that, for any null feature, its sampling
distribution is (asymptotically) symmetric about 0. In the moderate-dimensional
setting in which the ratio between the dimension (number of features) p and the
sample size n converges to a fixed value, we construct the mirror statistic
based on the maximum likelihood estimation. In the high-dimensional setting
where p is much larger than n, we use the debiased Lasso to build the mirror
statistic. Compared to the Benjamini-Hochberg procedure, which crucially relies
on the asymptotic normality of the Z statistic, the proposed methodology is
scale free as it only hinges on the symmetric property, thus is expected to be
more robust in finite-sample cases. Both simulation results and a real data
application show that the proposed methods are capable of controlling the FDR,
and are often more powerful than existing methods including the
Benjamini-Hochberg procedure and the knockoff filter.
