Order statistics of the early-type galaxy luminosity function
  We apply order statistics (OS) to the bright end ($M_r < -22$) of the
luminosity distribution of early-type galaxies spectroscopically identified in
the SDSS DR7 catalog. We calculate the typical OS quantities of this
distribution numerically, measuring the expectation value and variance of the
$k^{th}$ most luminous galaxy in a sample with cardinality $N$ over a large
ensemble of such samples. From these statistical quantities we explain why and
in what limit the $k^{th}$ most luminous galaxies can be used as standard
candles for cosmological studies.
  Since our sample contains all bright galaxies including the brightest cluster
galaxies (BCG), based on OS we argue that BCGs can be considered as statistical
extremes of a well-established Schechter luminosity distribution when galaxies
are binned by redshift and not cluster-by-cluster. We presume that the reason
behind this might be that luminous red ellipticals in galaxy clusters are \em
not random \em samples of an overall luminosity distribution but biased by the
fact that they are in a cluster containing the BCG. We show that a simple
statistical toy model can reproduce the well-known magnitude gap between the
BCG and the second brightest galaxy of the clusters.
