  High-dimensional sparse generalized linear models (GLMs) have emerged in the
setting that the number of samples and the dimension of variables are large,
and even the dimension of variables grows faster than the number of samples.
False discovery rate (FDR) control aims to identify some small number of
statistically significantly nonzero results after getting the sparse penalized
estimation of GLMs. Using the CLIME method for precision matrix estimations, we
construct the debiased-Lasso estimator and prove the asymptotical normality by
minimax-rate oracle inequalities for sparse GLMs. In practice, it is often
needed to accurately judge each regression coefficient's positivity and
negativity, which determines whether the predictor variable is positively or
negatively related to the response variable conditionally on the rest
variables. Using the debiased estimator, we establish multiple testing
procedures. Under mild conditions, we show that the proposed debiased
statistics can asymptotically control the directional (sign) FDR and
directional false discovery variables at a pre-specified significance level.
Moreover, it can be shown that our multiple testing procedure can approximately
achieve a statistical power of 1. We also extend our methods to the two-sample
problems and propose the two-sample test statistics. Under suitable conditions,
we can asymptotically achieve directional FDR control and directional FDV
control at the specified significance level for two-sample problems. Some
numerical simulations have successfully verified the FDR control effects of our
proposed testing procedures, which sometimes outperforms the classical knockoff
method.
