  Motivated by the simultaneous association analysis with the presence of
latent confounders, this paper studies the large-scale hypothesis testing
problem for the high-dimensional confounded linear models with both
non-asymptotic and asymptotic false discovery control. Such model covers a wide
range of practical settings where both the response and the predictors may be
confounded. In the presence of the high-dimensional predictors and the
unobservable confounders, the simultaneous inference with provable guarantees
becomes highly challenging, and the unknown strong dependence among the
confounded covariates makes the challenge even more pronounced. This paper
first introduces a decorrelating procedure that shrinks the confounding effect
and weakens the correlations among the predictors, then performs debiasing
under the decorrelated design based on some biased initial estimator. Following
that, an asymptotic normality result for the debiased estimator is established
and standardized test statistics are then constructed. Furthermore, a
simultaneous inference procedure is proposed to identify significant
associations, and both the finite-sample and asymptotic false discovery bounds
are provided. The non-asymptotic result is general and model-free, and is of
independent interest. We also prove that, under minimal signal strength
condition, all associations can be successfully detected with probability
tending to one. Simulation and real data studies are carried out to evaluate
the performance of the proposed approach and compare it with other competing
methods.

Methodology