Annu. Rev. Astron. Astrophys. 1997. 35: 101-136
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In their useful study, Landy & Szalay (1992) initiated the recent discussions on the general or "inhomogeneous" Malmquist bias and its correction, i.e. how to deal with the bias of the first kind in the case of a general space density distribution. True, they encountered some problems in the practical application of the formula derived in their paper. In fact, the basic problem was that they assumed in the beginning a distance indicator that has a zero bias of the second kind (which is unbiased at all distances) and then derived an expression for the bias of the first kind.

In Teerikorpi (1993) the problem of the general correction was discussed with explicit reference to direct and inverse TF relations, which served to clarify some of the points raised by Landy & Szalay (1992). It was noted that Malmquist's formula (Equation 3) was a general one, and in modern terms is best interpreted as applicable to the direct TF relation, for a constant value of p (Malmquist's "star class") and requires that the limiting magnitude (ml) = infty. In that case, the distribution of distance moduli N(µ) refers to all moduli that could be observed without any cutoff in the magnitudes.

Interestingly, Feast (1972) had already given a formula quite similar to Equation 3, but now with a quite different meaning for the distribution of N(µ): In his formulation, N(µ) must be regarded as the distribution of the derived distance moduli of the galaxies in the observed sample. Inspection of Feast's (1972) derivation reveals the implicit assumption that the bias of the second kind is zero, and the end result was the same as that of Landy & Szalay (1992). As the inverse TF relation distance moduli have the second bias = 0, one can conclude that Feast's and Landy & Szalay's variant of Malmquist's formula applies to the inverse TF distance moduli.

Landy & Szalay's (1992) paper gave rise to a burst of independent discussions. Similar conclusions as in Teerikorpi's (1993) paper on the general correction and the inverse TF relation were given by Feast himself (1994), Hendry & Simmons (1994) (see also Hudson 1994, Strauss & Willick 1995).

To reiterate, Malmquist's formula (Equation 3), with N(µ) refering to all distance moduli (mlim = infty) in the considered sky direction, applies to the direct TF moduli. In the case of the inverse TF relation, N(µ) is the observed distribution. Hence, for making corrections of the first kind for the direct distance moduli, one needs information on the true space density distribution of galaxies, and the selection function (magnitude limit) does not enter the problem. Corrections for the inverse moduli depend directly on the distribution of apparent magnitudes (distance moduli) in the sample and, hence, on the selection function. In this sense, the biases of the first and second kind for the direct and inverse distance moduli have curious complementary properties.

One might be content with the above conclusions and try to use the inverse relation in situations where the bias of first kind appears: The correction does not need the knowledge of the true space distribution. In practice, the high inhomogeneity of the local galaxy universe requires corrections for differing directions, which divides galaxy samples into small subsamples where the detailed behavior N(µ) is difficult to derive.

The general Malmquist bias for the inverse distance moduli, even in the case of a homogeneous space distribution, can be quite complicated. Because the observed N(µ) usually first increases, reaches a maximum, and then decreases to zero, the bias is first negative (too small distances), then goes through zero, and at large derived distances is positive (i.e. the distances are too long). Something like this is also expected simply because the average bias of the whole sample should be zero.

An important special case of the first Malmquist bias of the direct distance moduli is that of a homogeneous space distribution, which shifts standard candles in a Hubble diagram by a constant amount in magnitude and leaves the expected slope of 0.2 intact. In this manner, Soneira (1979) argued for the local linearity of the Hubble law, in contrast to a nonlinear redshift-distance relationship espoused by Segal, where he argues against debilitating bias effects.

It has been suggested that the distribution of galaxies at least up to some finite distance range could be fractal in nature, with fractal dimension D approx 2 (e.g. Di Nella et al 1996). In that case one might be willing to use, instead of the classical Malmquist correction (for D = 3 or homogeneity), Equation 5, which corresponds to the average density law proportional to rD-3. However, though the classical formula applies to every direction in a homogeneous universe, the deformed (D = 2) formula is hardly useful in any single direction of a fractal universe because of the strong inhomogeneities. Note however that in this case the argument by Soneira (1979) would be equally valid.

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