Effect of spectral index distribution on estimating the AGN radio
  luminosity function
  In this paper, we scrutinize the effect of spectral index distribution on
estimating the AGN (active galactic nucleus) radio luminosity function (RLF) by
a Monte Carlo method. We find that the traditional bivariate RLF estimators can
cause bias in varying degree. The bias is especially pronounced for the
flat-spectrum radio sources whose spectral index distribution is more
scattered. We believe that the bias is caused because the $K$-corrections
complicate the truncation boundary on the $L-z$ plane of the sample, but the
traditional bivariate RLF estimators have difficulty in dealing with this
boundary condition properly. We suggest that the spectral index distribution
should be incorporated into the RLF analysis process to obtain a robust
estimation. This drives the need for a trivariate function of the form
$\Phi(\alpha,z,L)$ which we show provides an accurate basis for measuring the
RLF.
