10.3 Meaningful probabilities

It is meaningful to estimate (and calculate) \[\begin{equation*} P_k(t) = \int_0^t S(s) \alpha_k(s) ds,\quad t > 0,\quad k = 1, 2, 3, \end{equation*}\] the probability to die from cause \(k\) before time \(t\). Note that \[\begin{equation*} S(t) + P_1(t) + P_2(t) + P_3(t) = 1 \texttt{ for all $t > 0$}. \end{equation*}\] Now, estimation is straightforward with the following estimators: \[\begin{equation*} \hat{P}_k(t) = \sum_{i:t_i \le t} \hat{S}(t_i-)\frac{d_i^{(k)}}{n_i} \end{equation*}\] See the function comp at the end of this chapter for how to write code for this estimation.