Estimating and testing the common mean direction of several von Mises-Fisher populations with known concentrations

In a situation with several VON MISES-FISHER (or Langevin) distributions with a common mean direction but different concentration parameters, we derive the conditional distribution of the weighted sample mean given the combined sample resultant length and the distribution of the combined sample resulatant length. Using these results we treat estiamtion and testing the mean of the distributions.