eQTL analyses aim to identify genetic loci that explain variation in gene expression. Studies including the Genotype-Tissue Expression (GTEx) Project have demonstrated that eQTL can vary across tissues. Thus, tissue-specific eQTL analyses are needed to understand cross-tissue similarities and difference in genetic regulatory architecture. Although many tissues are represented in GTEx, tissue ascertainment is incomplete, and expression data from less accessible tissues are partially missing. This missingness reduces power for tissue-specific eQTL analyses. We propose a new method, Bivariate Normal Regression via Expectation Maximization (BNEM), for borrowing information across correlated tissues in the presence of missing data. Our approach leverages the larger sample size available in a thoroughly-ascertained surrogate tissue to draw more precise inferences on the genetic effects in a poorly-ascertained target tissue. Through extensive simulations, we verify that BNEM improves power while maintaining the type I error. We apply BNEM to eQTL mapping in 13 subtypes of brain tissue from GTEx, using expression in whole blood as the surrogate.