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Astron. Astrophys. 343, 713-719 (1999)

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8. The calibrator sample bias when one uses relevant field slope [FORMULA]

In the above discussion it is noteworthy that one does not have to know the intrinsic slope [FORMULA] for the calibrators. However, the calibrator sample must fulfill a special condition. It must be equivalent to a volume-limited sample, or reflect the cosmic Gaussian distribution, as was already noted in Teerikorpi (1990). From Eq. (23), it is clear that two calibrator samples with different [FORMULA] yield different zero-points [FORMULA] for the same [FORMULA] of the field galaxies, while there can exist only one such correct [FORMULA], the one corresponding to [FORMULA] or [FORMULA].

If [FORMULA] of the calibrator sample deviates from the cosmic [FORMULA], there is a systematical error in [FORMULA], resulting in an average error in the derived [FORMULA]:

[EQUATION]

or in terms of the average p:

[EQUATION]

This source of error is very significant if the calibrator sample has not been constructed with the intention of reaching the cosmic distribution of p, if for example the selection of galaxies has aimed at detection of Cepheids. In this way, one may have a volume-limited sample for each p (c.f. Theureau et al. 1997a), but this does not guarantee that the distribution ofpreflects the cosmic distribution from which the field galaxy sample has been drawn.

That this could be a genuine problem, consider the calibrator sample of KLUN with Cepheid distances. An indication of the calibrator sample bias is seen by comparing the distribution of calibrator p's with that of the whole sample. Because the latter is diameter limited, it suffers from a Malmquist bias, hence its median should be displaced to a larger value of p in comparison with the calibrators. Ekholm et al. (1999) show that there is no such shift, which implies that calibrators' [FORMULA] is larger than the cosmic value. Such a situation results in an overestimated [FORMULA] when one is compelled to use [FORMULA].

Application of Eqs. (25-26) requires a knowledge of the average [FORMULA] or [FORMULA] of the cosmic distribution functions to be compared with those of the calibrator sample. It is clear that any calculation of [FORMULA] from the field sample requires, besides a suitable method, the value of [FORMULA] itself, hence an iterative approach. Below we show that there is another much more convenient route which does not require an explicit knowledge of the difference [FORMULA] (or the value of [FORMULA]). However, it assumes a radial space distribution law of galaxy number density.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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