  The ability to accurately derive black hole (BH) masses at progressively
higher redshifts and over a wide range of continuum luminosities has become
indispensable in the era of large-area extragalactic spectroscopic surveys. In
this paper we present an extension of existing comparisons between rest-frame
UV and optical virial BH mass estimators to intermediate redshifts and
luminosities comparable to the local H$\beta$ reverberation mapped active
galactic nuclei (AGN). We focus on the MgII, CIV, and CIII] broad emission
lines and compare them to both H$\alpha$ and H$\beta$. We use newly acquired
near-infrared spectra from the FMOS instrument on the Subaru telescope for 89
broad-lined AGN at redshifts between 0.3 and 3.5, complemented by data from the
AGES survey. We employ two different prescriptions for measuring the emission
line widths and compare the results. We confirm that MgII shows a tight
correlation with H$\alpha$ and H$\beta$, with a scatter of ~0.25 dex. The CIV
and CIII] estimators, while showing larger scatter, are viable virial mass
estimators after accounting for a trend with the UV-to-optical luminosity
ratio. We find an intrinsic scatter of ~0.37 dex between Balmer and carbon
virial estimators by combining our dataset with previous high redshift
measurements. This updated comparison spans a total of 3 decades in BH mass. We
calculate a virial factor for CIV/CIII] logf(CIV/CIII])=0.87 with an estimated
systematic uncertainty of ~0.4 dex and find excellent agreement between the
local reverberation mapped AGN sample and our high-z sample.
