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

**3.1. Biases of the First and Second Kind**

In this review, I call these two problems Malmquist bias of the first and second kinds: The first kind is the general Malmquist bias, directly connected with the classical treatment by Malmquist (1920). One might briefly define these two aspects as follows:

- Malmquist bias of the first kind is the systematic average error in
the distance modulus
*µ*for a class of galaxies with "derived"*µ*=*µ*_{der}= constant. - Malmquist bias of the second kind is the systematic error in the
average derived <
*µ*_{der}> for the class of galaxies with "true"*µ*=*µ*_{true}= constant.

In discussions of the first bias, one is interested in the distribution of true distance moduli for the constant derived modulus, whereas the second bias is concerned with the distribution of derived distance moduli at a constant true distance (e.g. at a constant redshift, if the Hubble law is valid). These two biases have appeared under different names. Teerikorpi (1990, 1995) separated the study of the Hubble diagram into "distance against velocity" and "velocity against distance." Hendry & Simmons (1994) spoke about "Bayesian" and "frequentist" approaches in terms of mathematical statistics, whereas Willick (1994) emphasizes "inferred-distance problem" and "calibration problem." In his bias properties series, (e.g. Sandage 1994a) also made a clear difference between the classical Malmquist bias and the distance-dependent effect, and he showed in an illuminating manner the connection between the two.

More recently, Strauss & Willick (1995) have used the terms Malmquist bias and selection bias, whereby they emphasize by "selection" the availability of galaxies restricted by some limit (flux, magnitude, angular diameter). On the other hand, one may say that the first kind of Malmquist bias is also caused by a selection effect dictated by our fixed position in the universe and the distribution of galaxies around us, and in the special case of the inverse relation (Section 5), it also depends on the selection function.

The first kind of bias is closely related to the classical Malmquist (1920) bias; hence the name is well suited. However, Lyden-Bell (1992) prefers to speak about Eddington-Malmquist bias: Malmquist (1920) acknowledged that Eddington had given Equation 6, which corresponds to the special case of constant space density, before he had. As for the second bias, the situation is not so clear. However, in Malmquist (1922), in his Section II.4, one finds formally the equations needed for the calculation of the second bias, when the luminosity function has a gaussian distribution in M. Malmquist does not seem to have this application in mind, but in view of the widespread habit of speaking about the Malmquist bias (or Malmquist effect) in this connection, it might be appropriate to name the distance-dependent effect after him as well [another possibility would be to speak about Behr Behr's (1951) effect, cf Section 3.3].