Extending the Calibration of C iv-based Single-epoch Black Hole Mass Estimators for Active Galactic Nuclei^*

We used the multicomponent spectral decomposition code developed by Park et al. (2015) for modeling the Hβ region of our STIS data. In brief, the code works by first simultaneously fitting a pseudocontinuum that consists of a single power law, an Fe ii template, and a host-galaxy template in the surrounding continuum regions of 4430–4770 and $5080\mbox{--}5450\,\mathring{\rm{A}}$ and then fitting the Hβ emission line complex with Gauss–Hermite series functions (van der Marel & Franx 1993; Cappellari et al. 2002) for one broad emission component (Hβ) and three narrow emission components (Hβ and [O iii] $\lambda \lambda 4959,5007$) and two Gaussian functions for the nearby blended He ii $\lambda 4686$ emission line after subtracting the best-fit pseudocontinuum model (see Park et al. 2015 and references therein for details of the measurement procedure; see also Woo et al. 2006; Bennert et al. 2015; Runco et al. 2016). The Hβ line widths, ${\mathrm{FWHM}}_{{\rm{H}}\beta }$ and ${\sigma }_{{\rm{H}}\beta }$, are measured from the best-fit broad-line model (i.e., the Gauss–Hermite series function), and the continuum luminosity at 5100 Å, $\lambda {L}_{5100\mathring{\rm{A}} }$, is measured from the best-fit power-law model.

Note that there are two differences between the method adopted here and the approach given by Park et al. (2015), specifically in the model components used for the Fe ii emission and host-galaxy starlight. The template for host-galaxy starlight is excluded in this work because stellar absorption features, which are critical to achieving reliable host-galaxy template fits, are not observable in the small-aperture STIS spectra. The minimal contribution of host-galaxy light and the relatively low signal-to-noise ratio (S/N) and spectral resolution of the STIS optical data make it difficult to detect any host-galaxy features. Moreover, the fits did not converge when we included the host-galaxy starlight component in the model. As a rough check, we provided a crude estimate of an upper limit for host-galaxy starlight contribution to the STIS spectra using the object SBS 1116+583A, which shows the highest host-galaxy fraction in the ground-based spectroscopic observations (see Park et al. 2012b; Barth et al. 2015) from our STIS sample. The host-galaxy flux in the STIS spectrum can be roughly estimated by subtracting the AGN flux at 5100 Å, which is obtained by subtracting the HST imaging-based galaxy flux at 5100 Å (Bentz et al. 2013) from the ground-based spectroscopic total flux at 5100 Å (Park et al. 2012b), from the total flux at 5100 Å of the STIS spectrum. The resulting host-galaxy fraction in the STIS spectrum is found to be ∼31%. Note that the other AGNs will have much lower contributions due to the lower host-galaxy fractions shown by Park et al. (2012b) and Barth et al. (2015).

Available Fe ii templates for the Hβ region include empirically constructed monolithic templates by Boroson & Green (1992) and Véron-Cetty et al. (2004), a theoretical template by Bruhweiler & Verner (2008), and a semi-empirical multicomponent template by Kovačević et al. (2010). After performing extensive tests using each of the templates and a linear combination of them for our STIS data, we opted to use the template of Kovačević et al. (2010) based on its overall performance as quantified by the ${\chi }^{2}$-statistics and residuals of the fits (see also Barth et al. 2013, 2015). As expected, the multicomponent template generally performed better than the monolithic templates, particularly for the objects showing strong Fe ii emission. The Kovačević et al. (2010) template appears to be the best currently available for accurately fitting diverse Fe ii emission blends in AGNs by allowing for different relative intensities between five Fe ii multiplet subgroups. To sum up, we followed the method described by Park et al. (2015), except that we used the template of Kovačević et al. (2010) instead of that of Boroson & Green (1992) for Fe ii emission, and we omitted the host-galaxy starlight template from the fits.

For the Mg ii spectral region, we first fit a pseudocontinuum model in the surrounding continuum regions of 2450–2750 and 2850–3100 Å. The pseudocontinuum model is composed of a single power-law function representing the AGN featureless continuum, an Fe ii emission template, and an empirical model for the Balmer continuum. We adopted the UV Fe ii template, which is made from observations of I Zw 1, from Tsuzuki et al. (2006). Using the template of Tsuzuki et al. (2006) is arguably better for modeling the Mg ii line region than using that of Vestergaard & Wilkes (2001) because it contains semi-empirically constrained Fe ii contribution underneath the Mg ii line, while the template by Vestergaard & Wilkes (2001) has no Fe ii flux at all under the Mg ii line due to the difficulty of decomposing this spectral region.

Based on the investigations of Grandi (1982) and Wills et al. (1985; see also Malkan & Sargent 1982 for the first practical measurement of the Balmer continuum shape), Dietrich et al. (2002, 2003) described a practical procedure for Balmer continuum modeling in high-z quasar spectra that has become a standard practice for fitting the Balmer continuum. This empirical model assumes that the Balmer continuum is generated from partially optically thick gas clouds with a uniform effective temperature (${T}_{e}=15,\,000\,{\rm{K}}$) as

$$\begin{eqnarray}&&{F}_{\lambda }^{\mathrm{BaC}}(A,{T}_{e},{\tau }_{\mathrm{BE}})={{AB}}_{\lambda }({T}_{e})(1-{e}^{-{\tau }_{\mathrm{BE}}{(\lambda /{\lambda }_{\mathrm{BE}})}^{3}}),\lambda \leqslant {\lambda }_{\mathrm{BE}},\end{eqnarray} \tag{ 1 }$$

where A and ${\tau }_{\mathrm{BE}}$ are the normalized flux density and optical depth at the Balmer edge (${\lambda }_{\mathrm{BE}}=3646$ Å), respectively, and ${B}_{\lambda }({T}_{e})$ is the Planck function at the electron temperature T_e. At $\lambda \gt {\lambda }_{\mathrm{BE}}$, higher-order Balmer lines using the relative intensity calculations from Storey & Hummer (1995) are used to represent the smooth rise to the Balmer edge. Many studies (e.g., Kurk et al. 2007; Wang et al. 2009; Greene et al. 2010; De Rosa et al. 2011, 2014; Ho et al. 2012; Shen & Liu 2012; Kokubo et al. 2014) have used variants of this method with slightly different ways of constraining the model parameter ranges based on the available data quality and spectral coverage. We found that if we treated all three parameters in the model ($A,{T}_{e},{\tau }_{\mathrm{BE}}$) as free parameters during fitting, as was done by Wang et al. (2009) and Shen & Liu (2012), they were very poorly constrained due to the degeneracy with the power-law continuum and Fe ii emission blends.

Recently, Kovačević et al. (2014) suggested an improved way to constrain the normalization A: by taking into account the fact that A can be obtained by calculating the sum of all intensities of higher-order Balmer lines at the Balmer edge. We followed this procedure with slight modifications. The value of A was separately determined from the intensity calculation using high-order Balmer lines (up to 400) with the template line profile adopted from the best-fit Hβ emission-line model obtained above. The other parameters, T_e and ${\tau }_{\mathrm{BE}}$, were then fitted simultaneously with the Fe ii template and power-law function during pseudocontinuum modeling. However, it should be noted that, following the Dietrich approach, the temperature was finally fixed to be 15,000 K, and the optical depth was allowed to vary between 0.1 and 2. We also independently checked that the constrained Balmer continuum component only exhibited marginal changes over temperatures ranging from 10,000 to 30,000 K and optical depths varying from 0.1 to 2. Note that the resulting continuum luminosity estimates are consistent with each other within ∼0.04 dex scatter.

After subtracting the best-fit pseudocontinuum model, the Mg ii emission line was fitted using a linear combination of a sixth-order Gauss–Hermite series and a single Gaussian function to account for its full line profile, typically showing a more peaky core (i.e., narrower and sharper line peak) and more extended wings than a Gaussian profile, in the spectral region ∼2700–2900 Å. We use the full line profile without a decomposition of narrow and broad components for the line width measurements for UV lines in this work (the same approach adopted by P13), in contrast to Hβ. This is because no reliable and clear distinction between broad and narrow components in the UV lines is usually possible, and sometimes no narrow components of the UV lines are seen at all; their presence is still uncertain and under debate. Thus, the Mg ii line widths, ${\mathrm{FWHM}}_{\mathrm{Mg}{\rm{II}}}$ and ${\sigma }_{\mathrm{Mg}{\rm{II}}}$, are measured from the best-fit full line profile, and continuum luminosity at 3000 Å, $\lambda {L}_{3000\mathring{\rm{A}} }$, is measured from the best-fit power-law function. During fitting, Galactic absorption lines such as Fe ii $\lambda \lambda 2586,2600$, Mg ii $\lambda \lambda 2796,2803$, and Mg i $\lambda 2852$ (cf. Savaglio et al. 2004) are masked out with exclusion windows.

Spectral measurements for the C iv line region in the archival sample of the local 25 RM AGNs were described by P13. Here, we focus on analysis of the six objects with newly obtained STIS data. We used the same methods as in P13 for consistency and to avoid additional systematic biases. We fit the AGN featureless continuum with a single power-law function, and we chose to omit a UV iron template (e.g., Vestergaard & Wilkes 2001) from the fits because no clear contribution of iron emission over the C iv region was observed. Although we performed a test by including the UV iron template in the model as in P13, its contribution was too small to be constrained accurately with the template, at least in our sample (see also Shen et al. 2008, 2011).

After the best-fit continuum model was subtracted, the C iv emission line was fitted with a linear combination of a sixth-order Gauss–Hermite series and a single Gaussian function. The contaminating nearby blended emission lines (e.g., N iv] $\lambda 1486$, Si ii $\lambda 1531$, He ii $\lambda 1640$, and O iii] $\lambda 1663$) were fitted simultaneously as well using up to two Gaussian functions for each line. Again, we used the combined model of one Gauss–Hermite function and one Gaussian function to fit the full line profile of C iv without decomposing it into broad and narrow components. The C iv line widths, FWHM_{C iv} and σ_{C iv}, were measured from the best-fit full line profile, and continuum luminosity at 1350 Å, $\lambda {L}_{1350\mathring{\rm{A}} }$, was measured from the best-fit power-law function. Narrow absorption spikes were masked out using a $3\sigma$ clipping threshold during fitting, and broad absorption features around the line center were masked out manually with exclusion windows (see P13 and references therein for more details of the C iv measurement method and results for the archival sample).

Uncertainties for the above spectral measurements are estimated with the Monte Carlo method used by Park et al. (2012b) and P13 (see also Shen et al. 2011; Shen & Liu 2012). For each spectral region, 1000 mock spectra are generated by resampling the original spectra with the addition of Gaussian random noise based on the error spectrum for each object. We then measure line widths and luminosities from each of the mock spectra using the same measurement methods and take the standard deviation of the distribution of the measurements as the estimate of the measurement uncertainty.

Typical uncertainty levels of line widths for all objects are found to be ∼2%–4% with a maximum of ∼17% due to the high quality of the HST spectra. For continuum luminosity, we derive uncertainties of ∼1%–2% with a maximum of ∼6%. These are small compared to the overall systematic mass uncertainty of ∼0.4 dex in the SE virial method. Covariances between the measurement uncertainties of the line widths and luminosities for each object in a logarithmic scale are also estimated from the resulting distributions of the Monte Carlo simulations, which are given as $\mathrm{cov}(\mathrm{log}\lambda {L}_{\lambda },\mathrm{logFWHM})=\rho \sigma (\mathrm{log}\lambda {L}_{\lambda })\sigma (\mathrm{logFWHM})$ and $\mathrm{cov}(\mathrm{log}\lambda {L}_{\lambda },\mathrm{log}{\sigma }_{\mathrm{line}})=\rho \sigma (\mathrm{log}\lambda {L}_{\lambda })\sigma (\mathrm{log}{\sigma }_{\mathrm{line}})$, where $\mathrm{cov}$, ρ, and σ are the covariance, correlation coefficient, and measurement uncertainty of the logarithms of the luminosities and line widths, respectively. Table 3 lists the line widths and luminosities for our sample, along with the measurement uncertainties and their error correlation coefficients.

Table 3.  Ultraviolet Spectral Properties from C iv SE Estimates

Object	Telescope/Instrument	Date Observed	S/N	$E(B-V)$	$\mathrm{log}(\lambda {L}_{\lambda }/$erg s−1)	${\mathrm{FWHM}}_{\mathrm{SE}}$	${\sigma }_{\mathrm{SE}}$	${\mathrm{MAD}}_{\mathrm{SE}}$	$\rho (W,X)$	$\rho (W,Y)$	$\rho (W,Z)$
			(1450 or 1700 Å)		(1350 Å)	(C iv)	(C iv)	(C iv)
			(pix−1)	(mag)		(km s−1)	(km s−1)	(km s−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
Sample Presented in P13 a
3C 120	IUE/SWP	1994 Feb 19, 27; 1994 Mar 11	12	0.263	44.399 ± 0.021	3093 ± 291	3106 ± 157	2280 ± 74	−0.05	−0.35	−0.50
3C 390.3	HST/FOS	1996 Mar 31	18	0.063	43.869 ± 0.003	5645 ± 202	6154 ± 65	4674 ± 42	−0.11	−0.30	−0.32
Ark 120	HST/FOS	1995 Jul 29	17	0.114	44.400 ± 0.005	3471 ± 108	3219 ± 53	2345 ± 31	0.01	−0.30	−0.37
Fairall 9	HST/FOS	1993 Jan 22	24	0.023	44.442 ± 0.004	2649 ± 77	2694 ± 20	1987 ± 13	0.04	−0.21	−0.17
Mrk 279	HST/COS	2011 Jun 27	9	0.014	43.082 ± 0.004	4093 ± 388	2973 ± 53	2202 ± 18	0.01	−0.04	−0.11
Mrk 290	HST/COS	2009 Oct 28	24	0.014	43.611 ± 0.002	2052 ± 36	3531 ± 32	2544 ± 14	−0.00	−0.13	−0.20
Mrk 335	HST/COS	2009 Oct 31; 2010 Feb 08	29	0.032	43.953 ± 0.001	1772 ± 14	1876 ± 12	1311 ± 7	0.03	−0.02	0.02
Mrk 509	HST/COS	2009 Dec 10, 11	107	0.051	44.675 ± 0.001	3872 ± 18	3568 ± 9	2601 ± 6	0.02	−0.07	−0.03
Mrk 590	IUE/SWP	1991 Jan 14	17	0.033	44.094 ± 0.007	5362 ± 266	3479 ± 165	2630 ± 139	−0.09	−0.12	−0.05
Mrk 817	HST/COS	2009 Aug 04; 2009 Dec 28	38	0.006	44.326 ± 0.001	4580 ± 48	3692 ± 23	2756 ± 14	−0.01	−0.25	−0.21
NGC 3516	HST/COS	2010 Oct 04; 2011 Jan 22	20	0.038	42.615 ± 0.002	2658 ± 34	4006 ± 49	2864 ± 29	−0.03	0.07	0.06
NGC 3783	HST/COS	2011 May 26	29	0.105	43.400 ± 0.001	2656 ± 444	2774 ± 91	2014 ± 9	−0.01	−0.01	−0.17
NGC 4593	HST/STIS	2002 Jun 23, 24	10	0.022	43.761 ± 0.005	2952 ± 166	2946 ± 162	2135 ± 33	−0.01	−0.00	−0.02
NGC 5548	HST/COS	2011 Jun 16, 17	36	0.018	43.822 ± 0.001	1785 ± 82	4772 ± 80	3528 ± 102	0.02	−0.11	−0.02
NGC 7469	HST/COS	2010 Oct 16	32	0.061	43.909 ± 0.001	2725 ± 66	2849 ± 237	2060 ± 15	−0.02	−0.01	−0.14
PG 0026+129	HST/FOS	1994 Nov 27	25	0.063	45.236 ± 0.005	1604 ± 50	4965 ± 113	3610 ± 77	−0.04	−0.11	−0.22
PG 0052+251	HST/FOS	1993 Jul 22	21	0.042	45.292 ± 0.004	5380 ± 87	4648 ± 50	3463 ± 30	−0.12	−0.54	−0.61
PG 0804+761	HST/COS	2010 Jun 12	34	0.031	45.493 ± 0.001	3429 ± 23	2585 ± 20	1932 ± 13	0.04	0.14	0.17
PG 0953+414	HST/FOS	1991 Jun 18	18	0.012	45.629 ± 0.005	3021 ± 74	3448 ± 55	2472 ± 35	−0.02	−0.43	−0.48
PG 1226+023	HST/FOS	1991 Jan 14, 15	93	0.018	46.309 ± 0.001	3609 ± 29	3513 ± 29	2595 ± 19	−0.18	−0.52	−0.60
PG 1229+204	IUE/SWP	1982 May 01, 02	28	0.024	44.609 ± 0.009	4023 ± 163	2621 ± 90	1989 ± 62	−0.29	−0.48	−0.49
PG 1307+085	HST/FOS	1993 Jul 21	14	0.030	45.113 ± 0.006	3604 ± 111	4237 ± 80	3010 ± 54	−0.13	−0.57	−0.58
PG 1426+015	IUE/SWP	1985 Mar 01, 02	45	0.028	45.263 ± 0.004	4220 ± 258	4808 ± 305	3549 ± 95	−0.10	−0.23	−0.42
PG 1613+658	HST/COS	2010 Apr 08, 09, 10	37	0.023	45.488 ± 0.001	6398 ± 51	4204 ± 17	3286 ± 13	−0.01	−0.10	−0.04
PG 2130+099	HST/COS	2010 Oct 28	22	0.039	44.447 ± 0.001	2147 ± 18	2225 ± 47	1554 ± 21	0.02	−0.06	−0.07
New Sample Presented Here
Arp 151	HST/STIS	2013 Apr 29	6	0.012	41.791 ± 0.017	1489 ± 26	2900 ± 61	1864 ± 35	−0.03	−0.38	−0.46
Mrk 1310	HST/STIS	2013 Jun 07	5	0.027	41.715 ± 0.025	1434 ± 78	2447 ± 108	1603 ± 54	0.00	−0.25	−0.31
Mrk 50	HST/STIS	2012 Dec 12	19	0.015	43.213 ± 0.003	2807 ± 63	4443 ± 160	3140 ± 115	0.02	−0.13	−0.10
NGC 6814	HST/STIS	2013 May 07	6	0.164	41.105 ± 0.021	2651 ± 264	2804 ± 103	2096 ± 59	0.02	−0.07	−0.11
SBS 1116+583A	HST/STIS	2013 Jul 12	13	0.010	42.867 ± 0.005	3253 ± 302	3315 ± 231	2302 ± 121	−0.04	−0.13	−0.15
Zw 229-015	HST/STIS	2013 Jul 23	17	0.064	43.129 ± 0.007	2573 ± 71	2608 ± 56	1891 ± 33	−0.05	−0.20	−0.18

Note. Col. (1) Name. Col. (2) Telescope/instrument from which archival UV spectra were obtained. Note that the new COS spectra were obtained after 2009. Col. (3) Observation date for combined spectra. Col. (4) S/N per pixel at 1450 or 1700 Å in the rest frame. Col. (5) Value of $E(B-V)$ from the NED based on the recalibration of Schlafly & Finkbeiner (2011). Col. (6) Continuum luminosity measured at 1350 Å. Col. (7) FWHM measured from SE spectra. Col. (8) Line dispersion (${\sigma }_{\mathrm{line}}$) measured from SE spectra. Col. (9) MAD (mean absolute deviation around weighted median) measured from SE spectra. Col. (10) Correlation coefficient between measurement errors of W and X, where $W=\mathrm{log}\lambda {L}_{\lambda }$ at 1350 Å and $X={\mathrm{logFWHM}}_{\mathrm{SE}}$. Col. (11) Correlation coefficient between measurement errors of W and Y, where $Y=\mathrm{log}{\sigma }_{\mathrm{SE}}$. Col. (12) Correlation coefficient between measurement errors of W and Z, where $Z={\mathrm{logMAD}}_{\mathrm{SE}}$.

^aNote that the sample and measurements are from P13. One difference here is that measurements for MAD and error correlations have been included.

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There are several issues in regard to measuring continuum luminosities and line widths accurately. It is important to take into account the Balmer continuum over the Mg ii line region in order to accurately decompose the power-law continuum for the luminosity measurements. Based on our investigation of the STIS data, the $\lambda {L}_{3000\mathring{\rm{A}} }$ values will be overestimated by ∼0.14 dex, on average, if the Balmer continuum component is not accounted for. This is consistent with the investigation of Shen & Liu (2012), who found a ∼0.12 dex systematic offset. This bias will then be propagated into the final ${M}_{\mathrm{BH}}$ estimates by as much as a ∼0.07 dex (∼17%) systematic offset if the Balmer continuum model is not included properly.

If the original Vestergaard & Wilkes (2001) Fe ii template is used, the FWHM (${\sigma }_{\mathrm{line}}$) estimates are overestimated by ∼0.03 (0.07) dex, on average, compared with the results derived using the template of Tsuzuki et al. (2006). This will again be propagated into the ${M}_{\mathrm{BH}}$ estimates by up to ∼0.06 (0.14) dex of systematic offset, which is consistent with the result of Nobuta et al. (2012).

To make the maximum use of the wide spectral coverage of our STIS data, we also performed extensive tests of global continuum fits covering C iv to Hβ simultaneously. Using a more flexible double power-law model to represent the AGN featureless continuum, we fit a pseudocontinuum model including the Balmer continuum model and Fe ii emission to many line-free continuum regions (see also, e.g., Shen & Liu 2012 and Mejía-Restrepo et al. 2016 for related recent work). The global continuum fits produced results consistent with the local continuum fits described in the previous sections, except that the global fits failed to constrain the Fe ii emission on the red side of the Mg ii regions for Arp 151 and Mrk 1310. This is probably because the double power-law model is not flexible enough to properly describe the steep local slope changes around Mg ii for these two objects that are coming from intrinsic changes of spectral shapes and/or from strong internal reddening, along with the incompleteness of currently available UV Fe ii templates across the regions. In any case, there is no significant improvement of the global fits compared to the local fits.

The simultaneous coverage of our STIS data also makes it possible to consistently compare the major UV and optical emission lines (C iv, Mg ii, and Hβ) without biases from intrinsic variability (Figure 3). The C iv profile shows, on average, more peaky cores with extended wings than those of the Mg ii and Hβ lines (see also Wills et al. 1993; Brotherton et al. 1994). There is no significant velocity offset between the line peaks of the three emission lines. Figure 4 compares FWHM and ${\sigma }_{\mathrm{line}}$ line width measurements among the emission lines. Although it is hard to draw a clear picture due to small-number statistics, we use only our STIS sample of six AGNs in order to perform a consistent comparison between the line widths in quasi-simultaneously observed data. We find that the FWHM of C iv is, on average, smaller than that of Hβ (and Mg ii), which may indicate that the FWHM of C iv is not a good proxy for virial BLR velocity, probably due to contamination from a nonreverberating C iv core component (Denney 2012). This contamination would be one of the biases correlated with Eigenvector 1 (EV1; Boroson & Green 1992), as discussed by Runnoe et al. (2013a, 2014) and Brotherton et al. (2015). They investigated and used the peak flux ratio of the $\lambda 1400$ feature to C iv as a UV indicator of EV1 to correct for the C iv-based BH masses. However, the interpretation is not straightforward. It could also be the case that the BLR geometry is different for the regions emitting these three lines, resulting in different individual virial factors (f) for each line (see also Runnoe et al. 2013b). By contrast, σ_{C iv} is, on average, larger than ${\sigma }_{{\rm{H}}\beta }$, which is consistent with the simple virial expectation of stratified BLR structure and the shorter reverberation lags of C iv (Peterson & Wandel 1999; Kollatschny 2003), thus corroborating the use of ${\sigma }_{\mathrm{line}}$ over FWHM for C iv-based BH mass estimates. More detailed intercomparisons and systematic investigation of multiline properties including more objects from the literature will be presented in a forthcoming paper.

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Figure 5 shows the results of the calibration of C iv-based SE BH mass estimators against the Hβ-based RM masses using the full sample of 31 local RM AGNs with Equation (2), while Figure 6 presents the corresponding marginal projections of each pair of parameters of interest with one-dimensional marginalized distributions from the full posterior distribution, from which parameter covariances are simply identifiable. We take the best-fit values and uncertainties of the parameters of interest ($\alpha ,\beta ,\gamma ,$ and ${\sigma }_{\mathrm{int}}$) from posterior median estimates and 68% posterior credible intervals, as recommended by Kelly (2007) and Hogg et al. (2010).

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In each panel of Figure 5, the two mass estimates (RM and SE) are fairly consistent. The overall scatter of the SE BH masses based on the calibrated equation using the C iv FWHM (${\sigma }_{\mathrm{line}}$) compared to the RM BH masses are at the level of 0.37 dex (0.33 dex). This indicates generally quite good consistency, given the unavoidable object-to-object scatter of the virial factor f (0.31 dex; Woo et al. 2010), since we are adopting a single ensemble average value of the f factor for all objects.

To assess the resulting model fit to the data, we present the posterior predictive distribution (blue shaded contour), which is generated from simulations according to the fit parameters of the model, to check whether the posterior prediction replicates the original observed data distribution reasonably well. As can be seen, the 95% credible region depicted by the light blue shaded contour describes most of the data distribution well, except for one outlier: NGC 6814. We have also calculated posterior predictive p-values by following the method of Chevallard & Charlot (2016; see also Gelman et al. 2013 and the PyMC User's Guide¹⁴ ). Using ${\chi }^{2}$ deviance as a discrepancy measure, the Bayesian p-value estimates are mostly ∼0.2, ranging from 0.14 to 0.26 in this work, indicating successful model fits to the data. Note that there is a problem (e.g., misfit or inadequacy in the descriptive model) if the p-value is extreme, i.e., $\lt 0.05$ or $\gt 0.95$. All the calibration results performed in this work for various cases are listed in Table 4. Note that the presence of the single outlier, which is not very extreme, does not alter any of the conclusions, and virtually the same result is obtained for the given uncertainties if the outlier is removed.

Table 4.  C iv ${M}_{\mathrm{BH}}$ Estimator Calibration Results $\mathrm{log}[{M}_{\mathrm{BH}}(\mathrm{RM})/{M}_{\odot }]=\alpha +\beta \mathrm{log}({L}_{1350\mathring{\rm{A}} }/{10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1})\,+\,\gamma \mathrm{log}[{\rm{\Delta }}{\rm{V}}({\rm{C}}\,{\rm{IV}})/1000\,\mathrm{km}\,{{\rm{s}}}^{-1}]$

${\rm{\Delta }}V({\rm{IV}})$	α	β	γ	${\sigma }_{\mathrm{int}}$	Mean Offset	1σ Scatter	References
					(dex)	(dex)
Previous Calibrations
${\sigma }_{\mathrm{line}}$	6.73 ± 0.01	0.53	2	0.33	⋯	⋯	VP06
FWHM	6.66 ± 0.01	0.53	2	0.36	⋯	⋯	VP06
${\sigma }_{\mathrm{line}}$	6.71 ± 0.07	0.50 ± 0.07	2	0.28 ± 0.04	0.00	0.295	P13
FWHM	7.48 ± 0.24	0.52 ± 0.09	0.56 ± 0.48	0.35 ± 0.05	0.00	0.347	P13
This Work
${\sigma }_{\mathrm{line}}$	${6.90}_{-0.34}^{+0.35}$	${0.44}_{-0.07}^{+0.07}$	${1.66}_{-0.66}^{+0.65}$	${0.12}_{-0.06}^{+0.09}$	0.01	0.33	⋯
FWHM	${\bf{7}}.{{\bf{54}}}_{-0.27}^{+0.26}$	${\bf{0}}.{{\bf{45}}}_{-0.08}^{+0.08}$	${\bf{0}}.{{\bf{50}}}_{-0.53}^{+0.55}$	${0.16}_{-0.08}^{+0.10}$	0.00	0.37	Best fita
MAD	${7.15}_{-0.25}^{+0.24}$	${0.42}_{-0.07}^{+0.07}$	${1.65}_{-0.62}^{+0.61}$	${0.12}_{-0.06}^{+0.09}$	0.00	0.33	⋯
This Work (Fixing $\gamma =2$)
${\sigma }_{\mathrm{line}}$	${\bf{6}}.{{\bf{73}}}_{-0.07}^{+0.07}$	${\bf{0}}.{{\bf{43}}}_{-0.06}^{+0.06}$	${\bf{2}}$	${0.12}_{-0.06}^{+0.09}$	0.01	0.33	Best fita
FWHM	${6.84}_{-0.09}^{+0.09}$	${0.33}_{-0.07}^{+0.07}$	2	${0.22}_{-0.10}^{+0.11}$	−0.01	0.43	⋯
MAD	${\bf{7}}.{{\bf{01}}}_{-0.07}^{+0.07}$	${\bf{0}}.{{\bf{41}}}_{-0.06}^{+0.06}$	${\bf{2}}$	${0.12}_{-0.06}^{+0.09}$	0.00	0.33	Best fita
This Work (Fixing $\beta =0.5$)
${\sigma }_{\mathrm{line}}$	${6.99}_{-0.34}^{+0.34}$	0.5	${1.49}_{-0.62}^{+0.63}$	${0.12}_{-0.06}^{+0.09}$	0.01	0.34	⋯
FWHM	${7.62}_{-0.23}^{+0.23}$	0.5	${0.31}_{-0.45}^{+0.46}$	${0.16}_{-0.08}^{+0.10}$	0.01	0.38	⋯
MAD	${7.23}_{-0.24}^{+0.24}$	0.5	${1.41}_{-0.58}^{+0.57}$	${0.12}_{-0.06}^{+0.09}$	0.01	0.34	⋯
This Work (Fixing $\beta =0.5$ and $\gamma =2$)
${\sigma }_{\mathrm{line}}$	${6.72}_{-0.07}^{+0.07}$	0.5	2	${0.12}_{-0.06}^{+0.08}$	0.01	0.35	⋯
FWHM	${6.82}_{-0.09}^{+0.09}$	0.5	2	${0.26}_{-0.11}^{+0.11}$	0.00	0.47	⋯
MAD	${7.00}_{-0.07}^{+0.07}$	0.5	2	${0.12}_{-0.06}^{+0.09}$	0.01	0.35	⋯

Note. The mean offset and $1\sigma$ scatter for our calibrations are measured from the average and standard deviations of mass residuals between the RM masses and calibrated SE masses, ${\rm{\Delta }}=\mathrm{log}{M}_{\mathrm{BH}}(\mathrm{RM})-\mathrm{log}{M}_{\mathrm{BH}}(\mathrm{SE})$. Note that the apparent big difference in ${\sigma }_{\mathrm{int}}$ estimates between the previous calibrations and this work is mostly due to the differences in the adopted RM mass error and statistical model. The uncertainty of $\mathrm{log}f$ (i.e., 0.31 dex) is added in quadrature to the uncertainties of the RM BH masses in this work. The standard deviation (σ) of the t-distribution is by definition different (larger) from that of the Gaussian distribution due to the heavy tail when the degrees-of-freedom parameter is small. In this case, the ${\sigma }_{\mathrm{int}}$ parameter of the t-distribution model is not the same as the data spread (σ) of the t-distribution. Boldface font indicates bit-fit values.

^aWe suggest these calibrations as the best ${M}_{\mathrm{BH}}$ estimators.

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The luminosity slope, β, is consistent with the photoionization expectation (i.e., 0.5) within the uncertainties for both FWHM-based and ${\sigma }_{\mathrm{line}}$-based estimators. This may also imply that the size–luminosity relation for the UV 1350 Å continuum has a slope consistent with the relation for the optical 5100 Å continuum (e.g., ${0.533}_{-0.033}^{+0.035}$ by Bentz et al. 2013). The velocity slope, γ, for the ${\sigma }_{\mathrm{line}}$-based estimator is close to the virial expectation (i.e., 2) within the uncertainties, while this is not the case for the FWHM-based estimator, whose γ value is consistent with zero, given the uncertainty interval. This is generally consistent with the calibration result based on C iv FWHM by Shen & Liu (2012), who found a much smaller slope (0.242) than 2, although their luminosity dynamic range probed is much higher than ours. And the ${\sigma }_{\mathrm{line}}$-based SE masses show overall less intrinsic and total scatter than the FWHM-based masses. Thus, this indicates that the C iv σ is a better tracer of the BLR velocity field than the C iv FWHM because it is closer to the virial relation and shows less scatter in mass estimates. These results are in overall agreement with those of Denney (2012), who found that the C iv FWHM is much more affected by a contaminating nonvariable C iv core component (see also P13 and Denney et al. 2013 for related discussion and interpretation). Velocity slopes that are shallower (0.50, 1.66) than the virial expectation (2) would also be expected in part due to an additional dispersion of the measured line-of-sight velocities stemming from orientation dependence (see Shen & Ho 2014).

Our final best fits are as follows (see also Table 4):

$$\begin{eqnarray}\mathrm{log}\left[\displaystyle \frac{{M}_{\mathrm{BH}}(\mathrm{SE})}{{M}_{\odot }}\right] & = & {6.73}_{-0.07}^{+0.07}+{0.43}_{-0.06}^{+0.06}\,\mathrm{log}\left(\displaystyle \frac{\lambda {L}_{1350\mathring{\rm{A}} }}{{10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right)\\ & & +2\,\mathrm{log}\left[\displaystyle \frac{\sigma ({\rm{C}}\,{\rm{IV}})}{1000\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right],\end{eqnarray} \tag{ 3 }$$

with the overall scatter against RM masses of 0.33 dex, which is defined by the standard deviation of mass residuals ${\rm{\Delta }}=\mathrm{log}{M}_{\mathrm{BH}}(\mathrm{RM})-\mathrm{log}{M}_{\mathrm{BH}}(\mathrm{SE})$, and

$$\begin{eqnarray}\mathrm{log}\left[\displaystyle \frac{{M}_{\mathrm{BH}}(\mathrm{SE})}{{M}_{\odot }}\right] & = & {7.54}_{-0.27}^{+0.26}+{0.45}_{-0.08}^{+0.08}\,\mathrm{log}\left(\displaystyle \frac{\lambda {L}_{1350\mathring{\rm{A}} }}{{10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right)\\ & & +{0.50}_{-0.53}^{+0.55}\mathrm{log}\left[\displaystyle \frac{\mathrm{FWHM}\ ({\rm{C}}\,{\rm{IV}})}{1000\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right],\end{eqnarray} \tag{ 4 }$$

with the overall scatter against RM masses of 0.37 dex. For the case of the ${\sigma }_{\mathrm{line}}$-based estimator, note that the value of γ is fixed at 2 (i.e., consistent with the virial expectation) in our final analysis because it is consistent with 2 within the uncertainty estimate when we treat it as a free parameter. Fixing γ to the physically motivated value helps to avoid possible object-by-object biases and systematics due to small-number statistics and reduces the uncertainties of the other resulting regression parameters (see also P13). The overall scatter of the ${\sigma }_{\mathrm{line}}$-based calibration is virtually unchanged if we fix $\gamma =2$, while that of the FWHM-based calibration increases considerably if γ is fixed to 2 (see Table 4).

In Figure 7, we show the calibration results in a three-dimensional space of luminosity, velocity, and mass for clarity. There is a strong dependence of mass on luminosity, while there is a much weaker dependence of mass on velocity, partly due to the small dynamic range of broad-line velocity in our sample. Especially for FWHM, the change of mass as a function of velocity is only marginal. As expected from the very small value of γ for the FWHM-based estimator, it seems that the C iv FWHM velocity characterization does not significantly add useful information to mass estimates, which is consistent with the result by Shen & Liu (2012; see also discussions on the importance of line width information by Croom 2011 and Assef et al. 2012). It would thus be possible to achieve a comparable level of accuracy in mass estimates using the luminosity information only, at least for the present sample. This much-weaker dependence of FWHM on mass than that of ${\sigma }_{\mathrm{line}}$ can also be observed from the projected RM mass-SE velocity plane in Figure 8, which reinforces that ${\sigma }_{\mathrm{line}}$ is a better velocity width measurement for the C iv line than FWHM. We generally recommend using the ${\sigma }_{\mathrm{line}}$-based C iv SE ${M}_{\mathrm{BH}}$ estimator instead of the FWHM-based estimator, since it is closer to virial relation and shows a better correlation and less scatter against the RM masses.

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In this work (i.e., Equations (3) and (4) and Table 4), we have used the virial factor, $\mathrm{log}f=0.71$, taken from Park et al. (2012a) and Woo et al. (2010) for a consistent comparison with the result of P13. If one wants to use a more recent value of the virial factor for BH mass estimates, e.g., the recently updated virial factor, $\mathrm{log}f=0.65$, from Woo et al. (2015), one can simply subtract the difference (0.06) between the adopted virial factors from the normalization, α, of all the calibration results in this work. Our results can similarly be rescaled to any other adopted value for virial factor f.

The final best-fit calibrated equations (Equations (3) and (4)) are very similar to those of our previous work (see Equations (2) and (3) of P13). If we compare the two mass estimates based on both estimators using the same measurements of our sample, there are very small offsets (0.02–0.03 dex) with small scatters of ∼0.09 dex for both FWHM-based and ${\sigma }_{\mathrm{line}}$-based masses, which are mostly coming from slight differences in the slopes between P13 and this work. Although the changes in the adopted virial relation are modest, it is worth noting that this work directly extends the applicability of the C iv-based ${M}_{\mathrm{BH}}$ estimators toward lower BH masses ($\sim {10}^{6.5}$ M_☉) than were present in the P13 sample.

The advantages of the adopted statistical model using Stan in this work are (1) using the outlier-robust t-distribution as an alternative to the normal distribution for error distributions and (2) modeling the intrinsic distribution of covariates explicitly with a multivariate t-distribution. To check the performance of our model, we provide a comparison of results by performing the same regression work with other available methods (i.e., mlinmix_err and FITEXY).

Figure 9 compares the resulting posterior distributions of parameters obtained from three different regression methods for the same data (i.e., calibration of the FWHM-based estimator using the full sample of 31 local RM AGNs as obtained above). The Stan Bayesian model implemented for this work (Park 2017, in preparation) uses the Student t-distributions for measurement errors, intrinsic scatter, and the covariate distribution model. The mlinmix_err method, a Bayesian linear regression code developed by Kelly (2007), employs a normal mixture model for covariate distribution and assumes Gaussian distributions for measurement errors and intrinsic scatter. The FITEXY method, a widely used traditional ${\chi }^{2}$-based linear regression method (Tremaine et al. 2002; see also Park et al. 2012a and references therein), also uses Gaussian distributions for measurement errors and intrinsic scatter, but it has no model specified for the covariate distribution and does not take into account possible correlations between measurement errors.

The left panel in Figure 9 shows an overall consistency of results between the Bayesian methods, Stan, and mlinmix_err, except for the distributions of intrinsic scatter. The quite strong difference of the ${\sigma }_{\mathrm{int}}$ distributions is expected, because Stan uses t-distributed intrinsic scatter while mlinmix_err uses normally distributed intrinsic scatter. By definition, the ${\sigma }_{\mathrm{int}}$ of the t-distribution is smaller than that of the Gaussian distribution due to the broader tails of the t-distribution (see Kelly et al. 2012 and D. Park 2017, in preparation). Another noticeable difference between the posterior distributions is that the widths of the probability distributions for the regression parameters obtained from Stan are slightly wider than those from mlinmix_err. Although not significant, this seems to indicate more reliable uncertainty estimates with Stan, probably due to the flexibility of the adopted t-distributions with degrees-of-freedom parameters. Note that the t-distribution ranges widely from the Cauchy distribution to the normal distribution with a varying degrees-of-freedom parameter, but the number of Gaussian components for the normal mixture model used in mlinmix_err is fixed as a constant (e.g., 3 by default, although a few Gaussians are usually enough to obtain a reasonable description of the observed distributions of many astronomical samples and data, as is the case in this work).

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The right panel in Figure 9 also shows a overall consistency between the resulting distributions of the Bayesian Stan and the ${\chi }^{2}$-based FITEXY method. However, underestimates of the parameter uncertainties from FITEXY are a bit more noticeable, possibly due to the absence of the covariate model description and not accounting for correlations between measurement errors in FITEXY estimates. The parameter distributions from FITEXY are obtained with a bootstrapping method, so it may not be a consistent comparison with the Bayesian posterior distributions. Many zero values in the ${\sigma }_{\mathrm{int}}$ distribution of FITEXY are also noticeable, which indicates that many realizations of bootstrap samples are optimized without the addition of intrinsic scatter. This behavior is one of the downsides of the ${\chi }^{2}$-based FITEXY estimator, which employs a somewhat ad hoc iterative procedure to determine ${\sigma }_{\mathrm{int}}$ because it cannot be constrained simultaneously with the regression parameters (see Kelly 2011 and Park et al. 2012a).

The adopted best-fit parameters and uncertainties from the three methods are listed in Table 5 for comparison. Again, there is no significant difference between the parameter estimates; they are basically consistent with each other within the uncertainties. The primary reason for this consistency is that the measurement uncertainties for the covariates (line widths and luminosities) are very small in this work (i.e., only a few percent, on average, due to high-quality HST spectra). The resulting covariances between measurement errors are consequently very small as well, thus leading to virtually no effect of error correlations on the regression parameter estimates, even though there are correlations between measurement errors (see Table 3). To summarize, all three methods (Stan, mlinmix_err, and FITEXY) produce consistent results in this work, except for arguably more reliable parameter uncertainty estimates when using Stan, given the small measurement errors of the covariates. Although the difference is marginal in this work, more flexible t-distributed errors, as well as an explicit covariate model description, are generally recommended to get a correct central trend against outliers by avoiding the effects of possible unaccounted systematic errors (see Park 2017, in preparation, for details).

Table 5.  Comparing Calibration Results with Other Linear Regression Methods

Method	α	β	γ	${\sigma }_{\mathrm{int}}$	Mean Offset	1σ Scatter
					(dex)	(dex)
Stan (Bayesian)	${7.54}_{-0.27}^{+0.26}$	${0.45}_{-0.08}^{+0.08}$	${0.50}_{-0.53}^{+0.55}$	${0.16}_{-0.08}^{+0.10}$	0.00	0.37
mlinmix_err (Bayesian)	${7.51}_{-0.25}^{+0.25}$	${0.43}_{-0.07}^{+0.07}$	${0.57}_{-0.51}^{+0.50}$	${0.24}_{-0.09}^{+0.09}$	−0.00	0.37
FITEXY (${\chi }^{2}$-based)	7.50 ± 0.22	0.43 ± 0.08	0.59 ± 0.46	0.20 ± 0.10	−0.00	0.37

Note. For a consistent comparison, the exact same methodology of FITEXY used by P13 is applied.

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Although we prefer line dispersion (${\sigma }_{\mathrm{line}}$) to FWHM in measuring C iv line width, as investigated above, one downside of using ${\sigma }_{\mathrm{line}}$ is that it requires high-S/N data to accurately fit the line wings. Noisy data can lead to biases in line width measurements, especially when the line profile has very extended wings, as is typical of C iv lines (Denney et al. 2013; see also Fine et al. 2010).

Recently, Denney et al. (2016b) suggested another way of measuring line width: the MAD around the flux-weighted median wavelength. They suggested it as the most reliable method of line width measurement for low-quality data. The MAD is by definition less affected by the core and wing parts of the profile. Instead, the middle portions of the velocity profile (relative to the median velocity) contribute primarily to the determination of line width. The lower sensitivity of the MAD to the line core in comparison with FWHM is quite useful in order to obtain the least-biased line width measurement when there is a nonvarying core component in the C iv line profile (see Denney 2012). Such components are very hard to identify and remove without using multi-epoch RM data. Additionally, the MAD has the useful property of being less sensitive to high-velocity line wings than line dispersion (i.e., absolute deviation versus squared deviation as weights). This is important when using low-S/N data, which makes accurate characterization of line wings very difficult.

Thus, the MAD inherits some of the practical merits of both ${\sigma }_{\mathrm{line}}$ and FWHM and possibly works better in low-quality data. We have carried out MAD measurements for the broad lines in our sample, and we find good consistency between the MAD and ${\sigma }_{\mathrm{line}}$ measurements (Figure 10). The two measurements are very nicely correlated with a marginal scatter, while a poor correlation with a large scatter is observed between the FWHM and MAD. In this regard, the MAD may be the best line width measurement method for the C iv emission line when using survey-quality spectra, as advocated by Denney et al. (2016b).

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As it is for the case of ${\sigma }_{\mathrm{line}}$, the γ of the MAD is also consistent with 2 for the given uncertainty estimates if left as a free parameter (see Table 4). Fixing the virial slope to $\gamma =2$, we find the following best-fit calibration of the SE C iv mass estimator based on the MAD as the measure of C iv line width:

$$\begin{eqnarray}\mathrm{log}\left[\displaystyle \frac{{M}_{\mathrm{BH}}(\mathrm{SE})}{{M}_{\odot }}\right] & = & {7.01}_{-0.07}^{+0.07}+{0.41}_{-0.06}^{+0.06}\,\mathrm{log}\left(\displaystyle \frac{\lambda {L}_{1350\mathring{\rm{A}} }}{{10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right)\\ & & +2\,\mathrm{log}\left[\displaystyle \frac{\mathrm{MAD}({\rm{C}}\,{\rm{IV}})}{1000\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right].\end{eqnarray} \tag{ 5 }$$

 In this case, the overall scatter against RM masses is 0.33 dex. The resulting MAD-based calibration and posterior distributions, which are not shown here, are very similar to the ${\sigma }_{\mathrm{line}}$-based, except for a slight difference in the intercept α (see Table 4).

In our calibration sample (i.e., local RM AGNs), C iv blueshifts are basically insignificant (see Richards et al. 2011; Shen 2013), so our calibration based on the local RM AGNs is relatively free of possible biases stemming from the effect of a large blueshift. However, the applicability of this calibration to high-z quasars may be uncertain, because large C iv blueshifts are known to be common in high-z, high-luminosity quasars (see, e.g., Richards et al. 2002). Available C iv-based ${M}_{\mathrm{BH}}$ estimators have been used for measuring the BH masses of a statistical sample of such high-z AGNs, simply based on assumption and extrapolation without a direct test. The best way of investigating and possibly correcting for the effect of C iv blueshifts on BH mass estimates would be to use direct C iv RM data (see Denney 2012 for the case of local AGNs). The number of AGNs with direct C iv RM observations is, however, very limited, due to the major practical difficulties of obtaining RM measurements for high-z, high-luminosity AGNs and the difficulty of obtaining space-based UV monitoring data for low-z AGNs.

Instead, Coatman et al. (2016, 2017; see also Shen & Liu 2012) recently provided a new empirical correction to C iv FWHM-based BH mass estimators as a function of C iv blueshift by comparing SE C iv measurements to SE Hα measurements.

In Figure 11, we compare the overall distributions of C iv FWHM-based BH mass estimates as a function of C iv blueshift using the spectral measurements of DR9 BOSS quasars.¹⁵ The blue shaded contour presents BH masses computed from the blueshift-corrected formula from Coatman et al. (2017), while the red and green shaded contours show those calculated with the updated recipe in this work and the original VP06 equation, respectively. As can be seen, at a large blueshift (≳2000 km s⁻¹), our estimator, which does not take into account the C iv blueshift, produces a similar mass distribution to the blueshift-corrected distribution. Note that the overestimated BH masses from the VP06 estimator are reduced by correcting for the blueshift effect on the C iv FWHM (Coatman et al. 2017). Thus, our locally calibrated FWHM-based ${M}_{\mathrm{BH}}$ estimator is applicable to a sample of high-z, high-luminosity quasars with high C iv blueshifts (e.g., ≳2000 km s⁻¹), giving a consistent mass scale on average.

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At a small blueshfit range ($\sim 0\mbox{--}1000$ km s⁻¹), where our calibration sample is distributed, the overall mass scale from our estimator is smaller than those of VP06 and Coatman et al. (2017) in the high-mass regime ($\gtrsim {10}^{8.5}M/{M}_{\odot }$) and larger in the low-mass regime ($\lesssim {10}^{8.5}\,M/{M}_{\odot }$). This trend has been described in detail by P13. Arguably, our calibration in this work (and P13) has resulted in a better agreement (overall mass scale) with RM masses than that of VP06 in terms of intrinsic scatter and using the higher-quality data set of the updated sample, at least in the mass range (${10}^{6.5-9.1}\,M/{M}_{\odot }$) where the calibration has been performed. Note that neither our calibration nor that of VP06 (also Coatman et al. 2017, which is based on VP06's calibration) is directly confirmed in the very high-mass BH regime ($\gtrsim {10}^{9}\,M/{M}_{\odot }$, where most of the Coatman et al. sample is) due to a lack of high-mass RM AGNs in the calibration samples. All of these trends are shown in Figure 13 of Coatman et al. (2017).

It is also worth noting that the striking mass increase toward the negative blueshift from using the equation of Coatman et al. (2017) is obviously unreliable, as already discussed by Coatman et al. (2017), due to their insufficient dynamic range of blueshift. This C iv blueshift-corrected recipe should therefore not be used for objects with a negative C iv blueshift.

As shown by Coatman et al. (2016), the Hα line seems to also be systematically changing as a function of C iv blueshift, although the Hα line measurements may not be very accurate due to a very low S/N for the Hα spectral region (mostly $\lesssim 10$ per resolution element; see their Table 1). If this is true, calibrating the SE C iv line to the SE Hα (or Hβ) line, as done by Coatman et al. (2016), would be flawed. In other words, correcting the C iv FWHM as a function of blueshift against the Hα FWHM would still be biased, since the Hα FWHM is also correlated with the C iv blueshift. As an ultimate goal, calibrating the SE C iv mass estimators against direct C iv RM data (or indirectly against the RM Balmer line if C iv RM data is unavailable) for a much larger sample including high-luminosity, high-C iv blueshift AGNs will be the best way to improve the SE mass method. However, given the difficulty of obtaining many direct C iv RM measurements for both low- and high-z AGNs and determining accurate blueshifts (and systemic redshifts; see, e.g., Denney et al. 2016a; Shen et al. 2016a), our simple calibration of SE C iv-based ${M}_{\mathrm{BH}}$ estimators will still be useful when estimating BH masses from C iv observations of AGNs over a wide range of redshift and luminosity.

In Figure 12, we compare the Hβ RM-based BH masses to the C iv FWHM-based SE BH masses from the corrected prescriptions presented by Denney (2012, their Equation (1)) and Runnoe et al. (2013a, their Equation (3)). Note that we use our sample of the local RM AGNs, except for four objects (PG 0026+129, PG 0052+251, PG 1226+023, and PG 1307+085) that do not have enough spectral coverage to measure the $\lambda 1400$ feature (see Figure 1 of P13). The peak flux of the emission-line blend of Si iv + O iv] (i.e., the $\lambda 1400$ feature) has been measured by fitting it with a local power-law continuum and multi-Gaussian functions following the same method as Runnoe et al. (2013a; see also Shang et al. 2007). As a direct comparison, we also show the C iv FWHM-based masses using our new calibration, Equation (4).

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The SE BH masses using the C iv line shape (FWHM/${\sigma }_{\mathrm{line}}$)–based correction by Denney (2012) show an overall scatter of 0.39 dex, which is the same as that of our calibration. However, this corrected prescription is not practically useful because it requires a ${\sigma }_{\mathrm{line}}$ measurement, as well as FWHM, to obtain the shape measurement for the correction to C iv masses. One can use ${\sigma }_{\mathrm{line}}$ directly, if it is available, rather than using FWHM. The slightly larger scatter of 0.43 dex is observed for the case of the peak flux ratio ($\lambda 1400$/C iv)–based correction by Runnoe et al. (2013a). The effect of the correction is less pronounced for our sample, which is not surprising based on the investigation by Brotherton et al. (2015), who found that the peak flux ratios measured for the RM AGN sample did not correlate with the difference between the Hβ and C iv velocity widths. Our simple calibration is a useful practical tool in situations in which such additional measurements are not available.