Distances and absolute magnitudes from trigonometric parallaxes

In astrophysical applications, derived quantities like distances, absolute magnitudes and velocities are used instead of the observed quantities, such as parallaxes and proper motions. As the observed values are affected by random errors and selection effects, the estimates of the astrophysical quantities can be biased if a correct statistical treatment is not used. This paper presents and discusses different approaches to this problem. We first review the current knowledge of Hipparcos systematic and random errors, in particular small-scale correlations. Then, assuming Gaussian parallax errors and using examples from the recent Hipparcos literature, we show how random errors may be misinterpreted as systematic errors, or transformed into systematic errors. Finally we summarise how to get unbiased estimates of absolute magnitudes and distances, using either Bayesian or non-parametrical methods. These methods may be applied to get either mean quantities or individual estimates. In particular, we underline the notion of astrometrybased luminosity, which avoids the truncation biases and allows a full use of Hipparcos samples.