## 5.2 Notation

Suppose that an individual may experience \(K\) consecutive events at times \(T_1<T_2<\cdot\cdot\cdot<T_K=T\), which are measured from the start of the follow-up.

Here different methods are proposed to estimate **conditional survival probabilities** such as \(P(T_2 > y \mid T_1 > x)\) or \(P(T_2 > y \mid T_1 \leq x)\), where \(T_1\) and \(T_2\) are ordered event times of two successive events.

The proposed methods are all **based on the Kaplan-Meier** estimator and the ideas behind the proposed estimators can also be used to estimate more general functions involving **more than two successive event times**. However, for ease of presentation and without loss of generality, we take \(K=2\) in this section. The extension to \(K>2\) is straightforward.

Let \((T_{1},T_{2})\) be a pair of successive event times corresponding to two ordered (possibly consecutive) events measured from the start of the follow-up.

Let \(T=T_{2}\) denote the total time and assume that both \(T_1\) and \(T\) are observed subject to a (univariate) random right-censoring variable \(C\) assumed to be independent of \((T_1,T)\). Due to censoring, rather than \((T_1,T)\) we observe \((\widetilde T_{1},\Delta_1,\widetilde T,\Delta_2)\) where \(\widetilde T_{1}=\min (T_{1},C)\), \(\Delta_{1}=I(T_{1}\leq C)\), \(\widetilde T=\min (T,C)\), \(\Delta_{2}=I(T\leq C)\), where \(I(\cdot)\) is the indicator function. Let \((\widetilde T_{1i},\Delta_{1i},\widetilde T_i,\Delta_{2i})\), \(1\leq i\leq n\) be independent and identically distributed data with the same distribution as \((\widetilde T_{1},\Delta_1,\widetilde T,\Delta_2)\).