  Large-scale multiple testing with highly correlated test statistics arises
frequently in many scientific research. Incorporating correlation information
in estimating false discovery proportion has attracted increasing attention in
recent years. When the covariance matrix of test statistics is known, Fan, Han
& Gu (2012) provided a consistent estimate of False Discovery Proportion (FDP)
under arbitrary dependence structure. However, the covariance matrix is often
unknown in many applications and such dependence information has to be
estimated before estimating FDP (Efron, 2010). The estimation accuracy can
greatly affect the convergence result of FDP or even violate its consistency.
In the current paper, we provide methodological modification and theoretical
investigations for estimation of FDP with unknown covariance. First we develop
requirements for estimates of eigenvalues and eigenvectors such that we can
obtain a consistent estimate of FDP. Secondly we give conditions on the
dependence structures such that the estimate of FDP is consistent. Such
dependence structures include sparse covariance matrices, which have been
popularly considered in the contemporary random matrix theory. When data are
sampled from an approximate factor model, which encompasses most practical
situations, we provide a consistent estimate of FDP via exploiting this
specific dependence structure. The results are further demonstrated by
simulation studies and some real data applications.
