#### Estimation of infinite-dimensional phase-type distributions

###### Autor: Mogens Bladt Petersen

###### If X has a phase–type distribution and N is any positive discrete random variable, then we say that the distribution of X · N belongs to the class of NPH distributions. Such distributions preserve the tractability and generality of phase–type distributions (often allowing for explicit solutions to stochastic models and being dense in the class of distributions on the positive reals) but with a different tail behaviour which is basically dictated by the tail of N. We thereby gain a tool for specifying distributions with a “body” shaped by X and with a tail defined by N. After reviewing the construction and basic properties of distributions from the NPH class, we will consider the problem of their estimation. To this end we will employ the EM algorithm, using a similar method as for finite–dimensional phase–type distributions. We consider the the fitting of a NPH distribution to observed data, (left-,right and interval-) censored data, theoretical distributions, histograms, and a couple of examples.