38
In our research, we simply assume that Zi and Wi have the same value for the ith
subjects, i = 1,..., n. The proposed method also applies when Zi and Wi have different
values.
Note that, the observation of T (or C) means that C (or T) will not be observed
in the short term, and vice versa. Thus, mathematically, T and C can be treated as
competing risks. Zheng and Klein (1995) shows that the competing risks are sufficient
to identify the marginal survival functions and construct a suitable estimator. Here,
we developed two sets of equations for Ti and Ci, which will be used to study insurance
companies’ “bankruptcy” and “acquisition”, respectively, in both Chapters 4 and 6.
Since the proposed method is an extension of the classic Cox model, the basic
marginal functions from the Cox model are adopted, as shown below:
• Hazard functions for the event time Ti and the dependent censoring time Ci
are, respectively, assumed to be:
λi(t∖Zi,Wi) = λ0(i)exp(Z^), (3.1)
Ψi(t∖Zi,Wβ = ⅜(i)exp(I⅛), (3.2)
where λo(i) and ψo(t) are unspecified baseline hazard functions; and β and βc are
unknown parameters with respective dimensions p × 1 and q × 1. Note that although
only Zi or Wi appears on the right side of the equation, both Zi and Wi contribute
to β and ∕5c, estimates of β and βc. Therefore, it is more accurate to write the hazard
functions given both Z1 and Wi.