Estimation of the false discovery proportion with unknown dependence.

Large-scale multiple testing with correlated test statistics arises frequently in many scientific research. Incorporating correlation information in approximating false discovery proportion has attracted increasing attention in recent years. When the covariance matrix of test statistics is known, Fan, Han & Gu (2012) provided an accurate approximation of False Discovery Proportion (FDP) under arbitrary dependence structure and some sparsity assumption. However, the covariance matrix is often unknown in many applications and such dependence information has to be estimated before approximating FDP. The estimation accuracy can greatly affect FDP approximation. In the current paper, we aim to theoretically study the impact of unknown dependence on the testing procedure and establish a general framework such that FDP can be well approximated. The impacts of unknown dependence on approximating FDP are in the following two major aspects: through estimating eigenvalues/eigenvectors and through estimating marginal variances. To address the challenges in these two aspects, we firstly develop general requirements on estimates of eigenvalues and eigenvectors for a good approximation of FDP. We then give conditions on the structures of covariance matrices that satisfy such requirements. Such dependence structures include banded/sparse covariance matrices and (conditional) sparse precision matrices. Within this framework, we also consider a special example to illustrate our method where data are sampled from an approximate factor model, which encompasses most practical situations. We provide a good approximation of FDP via exploiting this specific dependence structure. The results are further generalized to the situation where the multivariate normality assumption is relaxed. Our results are demonstrated by simulation studies and some real data applications.