We describe a Bayesian approach to estimating quasar black hole mass functions (BHMF) when using the broad emission lines to estimate black hole mass. We show how using the broad line mass estimates in combination with statistical techniques develope
d for luminosity function estimation leads to statistically biased results. We derive the likelihood function for the BHMF based on the broad line mass estimates, and derive the posterior distribution for the BHMF, given the observed data. We develop our statistical approach for a flexible model where the BHMF is modelled as a mixture of Gaussian functions. Statistical inference is performed using markov chain monte carlo (MCMC) methods. Our method has the advantage that it is able to constrain the BHMF even beyond the survey detection limits at the adopted confidence level, accounts for measurement errors and the intrinsic uncertainty in broad line mass estimates, and provides a natural way of estimating the probability distribution of any quantities derived from the BHMF. We conclude by using our method to estimate the local active BHMF using the z < 0.5 Bright Quasar Survey sources. At z = 0.2, the quasar BHMF falls off approximately as a power law with slope ~ 2 for M_{BH} > 10^8. Our analysis implies that z < 0.5 broad line quasars have a typical Eddington ratio of ~ 0.4 and a dispersion in Eddington ratio of < 0.5 dex (abridged).
We describe a Bayesian approach to estimating luminosity functions. We derive the likelihood function and posterior probability distribution for the luminosity function, given the observed data, and we compare the Bayesian approach with maximum-likel
ihood by simulating sources from a Schechter function. For our simulations confidence intervals derived from bootstrapping the maximum-likelihood estimate can be too narrow, while confidence intervals derived from the Bayesian approach are valid. We develop our statistical approach for a flexible model where the luminosity function is modeled as a mixture of Gaussian functions. Statistical inference is performed using Markov chain Monte Carlo (MCMC) methods, and we describe a Metropolis-Hastings algorithm to perform the MCMC. The MCMC simulates random draws from the probability distribution of the luminosity function parameters, given the data, and we use a simulated data set to show how these random draws may be used to estimate the probability distribution for the luminosity function. In addition, we show how the MCMC output may be used to estimate the probability distribution of any quantities derived from the luminosity function, such as the peak in the space density of quasars. The Bayesian method we develop has the advantage that it is able to place accurate constraints on the luminosity function even beyond the survey detection limits, and that it provides a natural way of estimating the probability distribution of any quantities derived from the luminosity function, including those that rely on information beyond the survey detection limits.
We present Gemini near-infrared spectroscopic observations of six luminous quasars at z=5.8$sim$6.3. Five of them were observed using Gemini-South/GNIRS, which provides a simultaneous wavelength coverage of 0.9--2.5 $mu$m in cross dispersion mode. Th
e other source was observed in K band with Gemini-North/NIRI. We calculate line strengths for all detected emission lines and use their ratios to estimate gas metallicity in the broad-line regions of the quasars. The metallicity is found to be supersolar with a typical value of $sim$4 Z_{sun}, and a comparison with low-redshift observations shows no strong evolution in metallicity up to z$sim$6. The FeII/MgII ratio of the quasars is 4.9+/-1.4, consistent with low-redshift measurements. We estimate central BH masses of 10^9 to 10^{10} M_{sun} and Eddington luminosity ratios of order unity. We identify two MgII $lambdalambda$2796,2803 absorbers with rest equivalent width W_0^{lambda2796}>1 AA at 2.2<z<3 and three MgII absorbers with W_0^{lambda2796}>1.5 AA at z>3 in the spectra, with the two most distant absorbers at z=4.8668 and 4.8823, respectively. The redshift number densities (dN/dz) of MgII absorbers with W_0^{lambda2796}>1.5 AA are consistent with no cosmic evolution up to z>4.
