
372 Current Trends in Bayesian Methodology with Applications
time to failure can be characterized by the cumulative distribution function
F (·) or the hazard function h(·). By analogy, if X
n
is the sojourn time in
state J
n−1
= i before going to state J
n
= j, then we can characterize the
distribution o f X
n
by F
ij
or equivalently by the conditional hazard function,
h
ij
(·), for each i 6= j ∈ S, defined for s ≥ 0 as:
h
ij
(s)
.
= lim
∆s↓0
1
∆s
P (s ≤ X
n
< s + ∆s|J
n−1
= i, J
n
= j, X
n
≥ s).
Hence we can alternatively characterize the semi-Markov process by (φ, P, h).
Finally we note the relationship between the two processes under consider-
ation. In a semi-Markov proces s ...