Annu. Rev. Astron. Astrophys. 1997. 35: 101-136
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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 / sigmainv2 = 1 / sigmadir2 - 1 / sigmaM2, where sigma 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 Ho 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 Mo.

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 sigmaslope > | 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].

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