This work examines the effects of multiple parton radiation on proton-proton $W^{\pm }$ boson production at the Relativistic Heavy Ion Collider (RHIC), focusing on the ratio of charged lepton pseudorapidity differential cross sections to test the SU(2) flavor symmetry violation in the proton quark sea. I show that the measured ratio is marginally affected by the fiducial constraints imposed on the QCD recoil radiation by the STAR experiment, and that the ratio is also stable with respect to transverse momentum resummation effects. Hence, the STAR data on the $W^{+}/$W^{-}$ cross section ratio provides a robust discrimination of $\bar{d}(x)$ and $\bar{u}(x)$ parton distribution functions (PDFs) at momentum fractions of order 0.1, shown by comparing predictions with various PDF models and using the L2 sensitivity analysis and reweighting methods.