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We analyse several large samples of active galactic nuclei (AGNs) in order to establish the best tools required to study the evolution of black hole mass ( M  BH ) and normalized accretion rate ( L / L  Edd ). The data include spectra from the Sloan Digital Sky Survey, 2dF QSO Redshift survey and 2dF SDSS LRG And QSO survey public surveys at  z  &lt; 2, and a compilation of smaller samples with 0 &lt;  z  &lt; 5. We critically evaluate the usage of the Mg  ii  λ2798 and C  iv  λ1549 lines, and adjacent continuum bands, as estimators of  M  BH  and  L / L  Edd , by focusing on sources where one of these lines is observed together with Hβ. We present a new, luminosity‐dependent bolometric correction for the monochromatic luminosity at 3000 Å,  L  3000 , which is lower by a factor of ∼1.75 than those used in previous studies. We also re‐calibrate the use of  L  3000  as an indicator for the size of the broad emission‐line region ( R  BLR ) and find thatR   BLR    ∝   L   3000    0.62, in agreement with previous results. We find thatFWHM   (   Mg    i  i   )  ≃   FWHM   (  H  β  )    for all sources withFWHM    (   Mg    i  i   )   ≲  6000kms    −  1. Beyond this full width at half‐maximum (FWHM), the Mg  ii  line width seems to saturate. The spectral region of the Mg  ii  line can thus be used to reproduce Hβ‐based estimates of  M  BH  and  L / L  Edd , with negligible systematic differences and a scatter of ∼0.3 dex. The width of the C  iv  line, on the other hand, shows no correlation with either that of the Hβ or the Mg  ii  lines and we could not identify the reason for this discrepancy. The scatter of  M  BH (C  iv ), relative to  M  BH (Hβ), is of almost 0.5 dex. Moreover, 46 per cent of the sources haveFWHM   (  C    i  v  )  ≲   FWHM   (  H  β  )   , in contrast with the basic premise of the virial method, which predictsFWHM    (  C    i  v  )   /   FWHM    (  H  β  )   ≃    3.7, based on reverberation mapping experiments. This fundamental discrepancy cannot be corrected based on the continuum slope or any C  iv ‐related observable. Thus, the C  iv  line cannot be used to obtain precise estimates of  M  BH . We conclude by presenting the observed evolution of  M  BH  and  L / L  Edd  with cosmic epoch. The steep rise of  L / L  Edd  with redshift up to  z  ≃ 1 flattens towards the expected maximal value of  L / L  Edd  ≃ 1, with lower  M  BH  sources showing higher values of  L / L  Edd  at all redshifts. These trends will be further analysed in a forthcoming paper.

Black hole growth to z = 2 – I. Improved virial methods for measuring M BH and L / L Edd