A graphical analysis of the systematic error of classical binned methods
  in constructing luminosity functions
  The classical 1/Va and PC methods of constructing binned luminosity functions
(LFs) are revisited and compared by graphical analysis. Using both theoretical
analysis and illustration with an example, we show why the two methods give
different results for the bins which are crossed by the flux limit curves
$L=L_{lim}(z)$. Based on a combined sample simulated by a Monte Carlo method,
the estimate $\phi$ of two methods are compared with the input model LFs. The
two methods give identical and ideal estimate for the high luminosity points of
each redshift interval. However, for the low luminosity bins of all the
redshift intervals both methods give smaller estimate than the input model. We
conclude that once the LF is evolving with redshift, the classical binned
methods will unlikely give an ideal estimate over the total luminosity range.
Page & Carrera (2000) noticed that for objects close to the flux limit
$\phi_{1/V_{a}}$ nearly always to be too small. We believe this is due to the
arbitrary choosing of redshift and luminosity intervals. Because
$\phi_{1/V_{a}}$ is more sensitive to how the binning are chosen than
$\phi_{PC}$. We suggest a new binning method, which can improve the LFs
produced by the 1/Va method significantly, and also improve the LFs produced by
the PC methods. Our simulations show that after adopting this new binning, both
the 1/Va and PC methods have comparable results.
