Introduction

The chapter first presents a construction of a semi-Markov process X on its canonical probability space, the space of all sequences of elements of the state space. It is then supposed that the semi-Markov chain is not directly observed but that there is a second finite state process Y whose transitions depend on the state of the hidden process X.

In Chapter 8 we described a process ﹛(Tn, Zn)﹜ with

We also had

With this decomposition, the parameters of the model are, with and and these are to be estimated.

An alternative decomposition is

With this decomposition, the parameters of the model that are to be estimated are and. In fact we shall estimate this second decomposition.

The work of Ferguson (1980), Burge (1997), Burge and Karlin (1997), Bulla (2006), Bulla and Bulla (2006), Bulla et al. (2010), Gu#x00E9;don and Cocozza-Thivent (1990) and others have used the second specification but with Aj i not depending on m. In these works the model is simulated by first selecting X0 = Z0 according to an initial distribution. If X0 = ei, then a duration T1 − T0 is selected from the distribution given by \and then a change of state according to the distribution and so on. The specification of these authors is equivalent to assuming that (m) in the first formulation does not depend on When applying this restricted formulation, it is necessary to determine if the model is rich enough to model the application.

Some authors give parametric forms to. This reduces the number of parameters of the model, but a suitable parametric model will need to be justified for each application. See Levinson (1986a,b), Ramesh and Wilpon (1992) Gu#x00E9;don (1992, 1999, 2003, 2007) and Gu#x00E9;don and Cocozza-Thivent (1990). A good review of various approaches is Yu (1986) but we shall use different notation and provide some alternative estimates.