About the Malmquist bias in the determination of $H_{0}$ and of distances of galaxies

International audienceWe provide the mathematical framework which elucidates the way of using a Tully-Fisher (TF) like relation in the determination of the Hubble constant $H_0$, as well as for distances of galaxies. The methods related to the so-called Direct and Inverse TF Relations (herein DTF and ITF) are interpreted as maximum likelihood statistics. We show that, as long as the same model is used for the calibration of the TF relation and for the determination of $H_0$, we obtain a coherent Hubble's constant. The choice of the model is motivated by reasons of robustness of statistics, it depends on selection effects in observation which are present in the sample. The difference on the distance estimates when using either the ITF or the DTF model is only due to random fluctuations. It is interesting to point out that the DTF estimate does not depend on the luminosity distribution of sources. Both statistics show a correction for a bias, inadequately believed to be of Malmquist type. The repercussion of measurement errors, and additional selection effects are also analyze