On the Minimal Value in Maier's Property Concerning Phase-Type Distributions

Abstract

In this work we focus our attention on the minimal parameter satisfying Maier's property for the derivatives of a continuous phase-type (PH) distribution. Our main result is a sharp lower bound on this parameter in terms of the poles of the Laplace-Stieltjes transform of the distribution. This problem is also related to a conjecture posed by C.A. O'Cinneide (Conjecture 6) [13] , concerning a PH representation of a phase-type distribution with minimal norm. For a PH(2) distribution, we carry out a detailed study, showing that in fact, a minimalnorm representation can be found in this case, and this norm coincides with the minimal parameter in Maier's property.