*Astron. Astrophys. 343, 713-719 (1999)*
## 4. Analytic calculation of (log *H*) vs. *d*_{a}
Assuming Gaussian distributions and sharp cutoffs at
and
, it is easy to calculate the
expected run of vs. normalized
distance . For simplicity, we write
the inverse relation as
We use as shorthand notation for
the logarithm of the linear diameter. At any fixed
the dispersion of *p* is
.
Now the task is to calculate the bias in
at each
. This is done starting from the same
initial formula (28) of T84 which was used to show the absence of bias.
However, in this case one must add the selection function
:
where and
.
This formula gives for the sample galaxies with true (log) diameter
, the (log) error in the diameter
deduced by the inverse relation (using
). It is directly reflected to the
derived value of at constant
(i.e., at fixed normalized distance
).
With the present choice of the selection function (sharp upper and
lower cutoffs), and after a change of variable,
Eq. (4) becomes
Here the term can be evaluated in
terms of the error function :
where and
correspond to the cutoffs
and
via Eq. (5).
Inspection of the formula confirms the qualitative conclusions of
Sect. 2. Putting , so that only the
upper cutoff remains (and recalling that
), the bias is seen to be positive:
the diameters are underestimated, leading to underestimated distances,
hence to overestimated . Note that
this calculation does not tell us the average bias in the sample.
© European Southern Observatory (ESO) 1999
Online publication: March 1, 1999
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