Decision-Theoretic Estimation of the Offspring Mean in Mortal Branching Processes

A phase-type distribution is the distribution of a killing time in a finite-state Markov chain. This paper describes some conjectures concerning these distributions. Some partial proofs and other evidence are offered in support of the conjectures. Some of the conjectures concern the geometry of the set of phase-type distributions, while others concern the properties of their densities.