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Velocity offsets in the broad Balmer lines of quasars and their temporal variations serve as indirect evidence for bound supermassive black hole binaries (SBHBs) at sub-parsec separations. In this work, we test the SBHB hypothesis for 14 quasars with double-peaked broad emission lines using their long-term (14–41 yr) radial velocity curves. We improve on the previous work by (i) using elliptical instead of circular orbits for the SBHBs, (ii) adopting a statistical model for radial velocity jitter, (iii) employing a Markov chain Monte Carlo method to explore the orbital parameter space efficiently and build posterior distributions of physical parameters, and (iv) incorporating new observations. We determine empirically that jitter comprises approximately Gaussian distributed fluctuations about the smooth radial velocity curves that are larger than the measurement errors by factors of a few. We initially treat jitter by enlarging the effective error bars and then verify this approach via a variety of Gaussian process models for it. We find lower mass limits for the hypothesized SBHBs in the range 108–1011 M⊙. For seven objects, the SBHB scenario appears unlikely based on goodness-of-fit tests. For two additional objects, the minimum SBHB masses are unreasonably large (>1010 M⊙), strongly disfavouring the SBHB scenario. Using constraints on the orbital inclination angle (which requires some assumptions) makes the minimum masses of four more objects unreasonably large. We also cite physical and observational arguments against the SBHB hypothesis for nine objects. We conclude that the SBHB explanation is not the favoured explanation of double-peaked broad emission lines. [ABSTRACT FROM AUTHOR]