such findings, although a full investigation is beyond the scope of this paper [[[see also]{}]{} @Schulze11; @Volonteri11; @Salviander13].

We note that in our study of specific accretion rates [@Aird12 see Section \[sec:accdist\]] we did not measure $p(\lambda {\;|\;}{\mathcal{M}_*},z)$ for high accretion rate sources ($\log\lambda\gtrsim-1$). In fact, our study excluded broad-line AGNs and was limited to moderate luminosity sources (${L_\mathrm{X}}=10^{42-44}$[$\mathrm{erg\;s^{-1}}$]{}). We demonstrated in @Aird12 that the exclusion of broad-line AGNs did not significantly bias our measurement of the slope of the specific accretion rate distribution at $-3\lesssim\log\lambda\lesssim-1$. In fact, broad-line AGNs only constitute a significant fraction of the AGN population at high luminosities [${L_\mathrm{X}}\gtrsim10^{44}$ [$\mathrm{erg\;s^{-1}}$]{}, e.g. @Barger05; @Trump09b]. Measurements of $p(\lambda {\;|\;}{\mathcal{M}_*},z)$ at $\log\lambda\gtrsim-1$, including broad-line and luminous (${L_\mathrm{X}}>10^{44}$ [$\mathrm{erg\;s^{-1}}$]{}) AGN, [[ are vital to confirm our model relating AGNs and their host galaxies, our extrapolation of the accretion rate distribution to the highest accretion rates, and our interpretation of the XLF. Such studies present a number of difficulties due to the relative rarity of such sources, the strong biases that can affect the properties of the observed population, and the challenge of disentangling the properties of luminous AGNs and their host galaxies. ]{}]{}

[[ However, the high luminosity AGN population 