The Tully-Fisher like relations are based on an observed
linear correlation between the absolute magnitude M
(or similar quantity such as 
with D the
linear diameter)
and the log line-width distance indicator p of galaxies
(
for spirals and 
for ellipticals). They allow to estimate
distance and
radial peculiar velocity of an individual galaxy
from its measured apparent magnitude m
(or similar quantity such as 
with d the
apparent diameter),
parameter p
and redshift z.
In this section, we recall in mind the basic statistical model
describing these relations
(see Triay et al. 1994 or Rauzy & Triay 1996
for details).
Regardless of the distance of the galaxy, selection effects in
observation and measurement errors, the theoretical probability
density (pd) in the M-p plane reads:
The Direct (i.e. Forward) Tully-Fisher (DTF) relation
assumes that it exists a random variable
,
statistically independent of p such as the
theoretical pd of Eq.
(1 (click here)) rewrites:
where
fp(p)
is the distribution function of the variable p
in the M-p plane.
The random variable
of zero mean and dispersion 
accounts for
the intrinsic scatter about the DTF straight line 
of zero-point bD and slope aD.
The distance modulus 
of an object reads:
where m is the apparent magnitude of the galaxy and
r its distance in Mpc.
Regardless of measurement errors, the observed probability
density takes the following form:
where
is a selection function accounting for selection effects
in observation on m and p,
is the spatial distribution function of sources (along the line-of-sight)
and 
is the normalisation factor warranting 
.
Under the three following assumptions:
0) No measurement errors on m and p are
present. Particularly, corrections on galactic extinction
and on inclination effects are supposed valid.
1)
The function 
is gaussian.
) Galaxies are homogeneously distributed in space,
which implies
that, whatever the line-of-sight direction, the distance
modulus distribution function reads
with
.
generally adopted for the distance of a galaxy
with measured m and p reads as
follows (see Appendix A for details,
or Lynden-Bell et al. 1988;
Landy & Szalay 1992;
Triay et al. 1994):
accounts for a volume correction.
This unbiased
distance estimator
does not depend on the selection function
in m and p
and on the specific shape of the distribution function
fp(p).
Its accuracy 
is proportional to the distance
estimate of the galaxy (see Appendix A):
of the radial peculiar velocity of a galaxy
with measured m, p and z reads thus as follows:
.
The accuracy
of this radial peculiar velocity estimator
is:
