Statistical biases in stellar astronomy: the Malmquist bias revisited

A unified analytical treatment of the Malmquist bias is presented. Depending on the data under consideration and on the way the mean values of absolute magnitude are calculated, three different types of Malmquist bias appear: integral bias, and magnitude- and distance-dependent ones. An analytical consideration of the distance-dependent bias, previously studied in extragalactic astronomy, is given in terms of the trigonometric parallax. In a quantitative treatment of the Spaenhauer diagram, in which the derived absolute magnitude is plotted versus the true parallax, we developed an easy-to-use method for determination of the region unaffected by the bias ('unbiased plateau'). Considering spectroscopic distance indicators, we point out that for any magnitude-limited sample spectroscopic distance and parallax approach constant limits as the true distance increases. We draw some examples from luminosity calibrations of stellar classes. A brief outline is also given of a tentative course of investigations, anticipating future space astrometry missions.