RMS Parametric Survival Models

Here’s Claude AI’s answer.

Variance of MLE of Mean Survival Time (Exponential with Right Censoring)

Setup

Let the data be t_1, \ldots, t_n with censoring indicators \delta_i \in \{0,1\} (1 = event). Define:

• d = \sum_i \delta_i — total number of events
• T = \sum_i t_i — total exposure time (events + censored)

MLE of the rate parameter

The log-likelihood is \ell(\lambda) = d \log\lambda - \lambda T, giving:

\hat\lambda = \frac{d}{T}

Variance of \hat\lambda

The Fisher information is \mathcal{I}(\lambda) = d/\lambda^2, so:

\widehat{\text{Var}}(\hat\lambda) = \frac{\hat\lambda^2}{d} = \frac{d}{T^2}

Variance of \hat\mu = 1/\hat\lambda

By the delta method, with g(\lambda) = 1/\lambda and g'(\lambda) = -1/\lambda^2:

\widehat{\text{Var}}(\hat\mu) = \frac{\hat\mu^2}{d} = \frac{T^2}{d^3}

Key insight

Precision depends only on the number of events d, not sample size or censoring pattern. The coefficient of variation of \hat\mu is 1/\sqrt{d}.

A 95% CI is best formed on the log scale:

\exp\!\left(\log\hat\mu \pm \frac{1.96}{\sqrt{d}}\right)