on $L_{\rm AGN}(5100\,{\rm \AA})$; the red dotted line is the regression of $L_{\rm AGN}(5100\,{\rm \AA})$ on $L({\mbox{\rm H$\beta$}}_{\rm broad})$, which we use in equation (\[eq:LHbLAGN\]). The coefficients for both fits are given in Table 2.[]{data-label="Figure:LAGNLHb"}](Figure2.eps)

Line-Width Relationships {#section:hblinewidths}
------------------------

We now consider the use of  and  as proxies for [cf. @Collin06; @Wang19]. The left panel of Figure \[Figure:windowhbwidths\] shows the relationship between ${\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})$, the  line dispersion in the rms spectrum, and ${\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}),$ the  line dispersion in the mean spectrum. The relationship is nearly linear (slope $ = 1.085\pm0.045$) and the intrinsic scatter is small ($0.079$dex). The fit coefficients are given in the first row of Table 3.

![The relationship between  line dispersion in the rms ${\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})$ and mean ${\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}})$ spectra is shown on the left. The relationship between  line dispersion in the rms spectrum () and FWHM in the mean spectrum () is shown on the right. Blue filled circles are for the RMDB sample (Table A1) and open green triangles are for the SDSS sample (Table A3). The solid lines are best fits to equation (\[eq:powerlaw\]) with coefficients in Table 3. The short dashed and long dashed lines indicate the $\pm 