**Up:** Calibration of the

Hereafter, the correlation between *p* and
the velocity estimator given Eq. (20 (click here))
is calculated
assuming the following hypothesis:

- 3)
Along a line-of-sight, the distribution of radial peculiar velocities
*v*
does not depend on distances *r* and is a gaussian of
mean *u* and dispersion ,
i.e. .

Under assumptions
1, 2 and 3, the probability density
of Eq. (11 (click here)) rewrites:

In order to calculate each term involved in the covariance
of *p* and , i.e. , the velocity estimator
given Eq. (9 (click here)) is rewritten
in terms of variables *v*, and :

Since *u* is a constant,
.
Integrating over the variable *v* gives for
the three quantities ,
and
:

where properties
(A5 (click here)a) and
(A5 (click here)b) have been used.
Replacing by and using definition
of *B* given Eq. (9 (click here)),
Eqs. (B3 (click here), B4 (click here), B5 (click here))
expressed in terms of *p*, *m* and read:

where the constant *C* is defined as follows:

By using properties (A4 (click here))
and (A5 (click here)a), it is found that integral of Eq.
(B9 (click here)) vanishes. It thus implies that *C*=0,
,
,
and finally
.
This result does not depend on the shape
of the selection function in *m* and *p*,
on the theoretical distribution function *f*_{p}(*p*) of
the variable *p* and on the mean radial peculiar
velocity *u* along the line-of-sight.
For a sample of galaxies satisfying hypotheses
1, 2 and 3, it thus turns
out that the observable *p* is not correlated
with the velocity estimator
as long as the model parameters entering into
via Eq. (9 (click here)) are correct. On the other hand, if a wrong value
is assumed for one of these calibration parameters,
say for example that
*a*^{D}, *b*^{D}
and are correct but not the Hubble constant
,
the covariance between *p* and
gives:

which does not vanish since
existing selection effects on apparent magnitude *m* correlate
variables *p* and (i.e. for selected
galaxies, observable
*p* increases in average with the distance estimator
). This point motivates the use of a null-correlation
approach
in order to calibrate Tully-Fisher like relations.

**Up:** Calibration of the
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