The recently developed rerandomized inverse variance weighted (RIVW) estimator provides a simple and efficient framework to break the winner's curse in two-sample Mendelian randomization (MR). However, this method does not account for sample structure (e.g., residual population stratification and sample overlap), a common source of confounding in MR studies. Sample structure can not only distort SNP-exposure and SNP-outcome association estimates but also induce correlation between them, leading exposure-side instrument selection to propagate bias to the outcome side. To address this challenge, we propose the bivariate RIVW (BRIVW) estimator to simultaneously account for the winner's curse and sample structure. The BRIVW estimator extends the RIVW framework by modeling the joint distribution of SNP-exposure and SNP-outcome association estimates, first adjusting their covariance matrix via linkage disequilibrium score regression to account for sample structure and then applying randomized instrument selection and bivariate Rao-Blackwellization to obtain unbiased post-selection association estimates together with an estimator of their covariance matrix. Under mild conditions, we show that the BRIVW estimator is consistent and asymptotically normal. The finite-sample performance of the proposed estimator is evaluated through extensive simulations and real data analyses.

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