UC Riverside

This thesis discusses the process of extracting the longitudinal asymmetry,

$A_L^{W\pm}$, describing $W\rightarrow\mu$ production in forward kinematic

regimes. This asymmetry is used to constrain our understanding of the

polarized parton distribution functions characterizing $\bar{u}$ and $\bar{d}$

sea quarks in the proton. This asymmetry will be used to constrain the overall

contribution of the sea-quarks to the total proton spin. The asymmetry is

evaluated over the pseudorapidity range of the PHENIX Muon Arms, $2.1 <

|\eta|2.6$, for longitudinally polarized proton-proton collisions at 510 GeV

$\sqrt{s}$. In particular, I will discuss the statistical methods used to

characterize real muonic $W$ decays and the various background processes is

presented, including a discussion of likelihood event selection and the

Extended Unbinned Maximum Likelihood fit. These statistical methods serve

estimate the yields of $W$ muonic decays, which are used to calculate the

longitudinal asymmetry.