The name is absent

38

In our research, we simply assume that Z_i and W_i have the same value for the i^th
subjects, i = 1,..., n. The proposed method also applies when Z_i and W_i 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 T_i and C_i, 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 T_i and the dependent censoring time C_i
are, respectively, assumed to be:

λ_i(t∖Z_i,W_i) = λ₀(i)exp(Z^),                   (3.1)

Ψi(t∖Z_i,Wβ = ⅜(i)exp(I⅛),               (3.2)

where λo(i) and ψo(t) are unspecified baseline hazard functions; and β and β_c are
unknown parameters with respective dimensions p × 1 and q × 1. Note that although
only Z_i or W_i appears on the right side of the equation, both Z_i and W_i contribute
to β and ∕5_c, estimates of β and β_c. Therefore, it is more accurate to write the hazard
functions given both Z₁ and W_i.

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