0.5219          0.4243           0.4857
    ph.ecog    -0.3824     -0.3121      -0.3557         -0.3541          -0.4042
   $\sigma$     0.4639      0.8835       1.0286          0.4847           0.5360
   $\gamma$     0.3095      0.5051         –             0.4083             –
   $\delta$     0.6674        –            –               –                –
      AIC      538.2100    545.9405     563.8323    [**536.0556**]{}     545.0486

  : NCCTG Lung Cancer data: Maximum likelihood estimates, AIC (best value in bold).[]{data-label="table:LungMLE"}

It is sometimes of interest to obtain information about the remaining life of individual cancer patients. This information is used for future planning of health care, which is of financial and medical importance. Specifically, the probability that patient $i$ survives until time $t$, given that he/she was alive at time $t_i$ is given by, $$\begin{aligned}
\label{RemainingLife}
G(t\vert t_i;\bm{\theta}) = {\mathbb P}(T\leq t\vert T> t_i) = \dfrac{G(t;\bm{\theta})-G(t_i;\bm{\theta})}{1-G(t_i;\bm{\theta})},\,\,\, t\geq t_i,\end{aligned}$$ where $G$ is the distribution under the model of interest. For an AFT model, the parameter $\bm{\theta}$ contains both the regression parameters as well as the parameters of the distribution of the errors. The simplest way to obtain an estimator of this probability consists of plugging in the MLE of $\bm{\theta}$ in (\[RemainingLife\]). The 100(1-$\alpha$)% prediction interval [@H09] for a patient that survived until time $t_i$ is $[T_i^L,T_i^R]$, which satisfies $G(T_i^L\vert t_i;\bm{\theta})=\alpha_1$ and $G(T_i^R\vert t_i;\bm{\theta})=1-\alpha_2$, with $\alpha_1 + \alpha_2=\alpha$. In our application we choose $\alpha_1=\alpha_2=0.05$, and 