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We conduct a comprehensive analysis of the relationship between centralgalaxies and their host dark matter halos, as characterized by the stellarmass-halo mass (SM-HM) relation, with rigorous consideration of uncertainties.Our analysis focuses on results from the abundance matching technique, whichassumes that every dark matter halo or subhalo above a specific mass thresholdhosts one galaxy. We discuss the quantitative effects of uncertainties inobserved galaxy stellar mass functions (GSMFs) (including stellar massestimates and counting uncertainties), halo mass functions (including cosmologyand uncertainties from substructure), and the abundance matching technique usedto link galaxies to halos (including scatter in this connection). Our analysisresults in a robust estimate of the SM-HM relation and its evolution from z=0to z=4. The shape and evolution are well constrained for z < 1. The largestuncertainties at these redshifts are due to stellar mass estimates; however,failure to account for scatter in stellar masses at fixed halo mass can lead toerrors of similar magnitude in the SM-HM relation for central galaxies inmassive halos. We also investigate the SM-HM relation to z=4, although theshape of the relation at higher redshifts remains fairly unconstrained whenuncertainties are taken into account. These results will provide a powerfultool to inform galaxy evolution models. [Abridged]

A Comprehensive Analysis of Uncertainties Affecting the Stellar Mass - Halo Mass Relation for 0