Annu. Rev. Astron. Astrophys. 1997. 35:
101-136
Copyright © 1997 by . All rights reserved |

**5.2. Problems in the Use of the Inverse Relation**

In principle, the inverse TF relation seems to be
a good solution to the bias of the second kind, though the scatter in the
average distance modulus is larger than for the direct relation in the
unbiased plateau (1 /
_{inv}^{2}
= 1 /
_{dir}^{2}
- 1 /
_{M}^{2},
where
_{M} is the dispersion of the general luminosity function). In
practice, there are a few more serious problems.

It is essential that there should be no selection according to *p*,
say, working against distant, very broad HI-line galaxies. Also noteworthy
is that the calibrator slope for the inverse relation, derived from bright
nearby galaxies, is not necessarily the correct slope for distant galaxies.
This was shown in a concrete manner in the study of the Virgo cluster by
Fouqué et al
(1990).
If the magnitude or diameter measurements are less accurate for the distant
sample than for the calibrators, then the correct slope differs from the
calibrator slope. If one ignores this problem, the inverse relation will
give distances that are too small or a value of H_{o} that is
too high. Theoretically,
Teerikorpi (1990)
concluded that a solution is to use the slope obtained for the distant
sample. However, this requires that the general luminosity function is
symmetric around a mean value M_{o}.

The correct slope for the inverse relation is especially
important because the aim is to extend measurements at once to large
distances, i.e. to extreme values of m and *p*. A small error in
the slope causes
large errors at large distances. Of course, it is also important to have
the direct relation correct, but in any case, caution is required with
regard to the expected Malmquist bias of the second kind, whereas the
very absence of the bias is motivation to use the inverse relation.

Hendry & Simmons
(1994)
made numerical experiments in order to see what is needed of the calibrators
in order to produce the correct inverse slope. Adjusting their experiments
to the Mathewson et al
(1992a,
b)
data, they concluded that if the number of calibrators is less than 40,
the uncertainty
_{slope} >
| *direct* *slope* - *inverse slope*|. In other words,
with a small calibrator sample, we
may think that we use the inverse relation, whereas, in fact, we actually
have determined and therefore use the direct one.

Recently, Hendry & Simmons 1994, 1995 formulated the inverse TF distance estimator within the framework and language of mathematical statistics, which confirmed the earlier conclusions on its unbiased nature as regards the Malmquist bias of the second kind. Further discussion on the statistical properties of the inverse TF relation as a distance indicator may be found in Triay et al (1994, 1996) [see also Appendix of Sandage et al (1995) for illuminating notes].