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  "path": "/t/rms-parametric-survival-models/28766#post_4",
  "publishedAt": "2026-05-30T20:19:55.000Z",
  "site": "https://discourse.datamethods.org",
  "textContent": "Here’s Claude AI’s answer.\n\n## Variance of MLE of Mean Survival Time (Exponential with Right Censoring)\n\n### Setup\n\nLet the data be t_1, \\ldots, t_n with censoring indicators \\delta_i \\in \\\\{0,1\\\\} (1 = event). Define:\n\n  * d = \\sum_i \\delta_i — total number of events\n  * T = \\sum_i t_i — total exposure time (events + censored)\n\n\n\n### MLE of the rate parameter\n\nThe log-likelihood is \\ell(\\lambda) = d \\log\\lambda - \\lambda T, giving:\n\n\\hat\\lambda = \\frac{d}{T}\n\n### Variance of \\hat\\lambda\n\nThe Fisher information is \\mathcal{I}(\\lambda) = d/\\lambda^2, so:\n\n\\widehat{\\text{Var}}(\\hat\\lambda) = \\frac{\\hat\\lambda^2}{d} = \\frac{d}{T^2}\n\n### Variance of \\hat\\mu = 1/\\hat\\lambda\n\nBy the delta method, with g(\\lambda) = 1/\\lambda and g'(\\lambda) = -1/\\lambda^2:\n\n\\widehat{\\text{Var}}(\\hat\\mu) = \\frac{\\hat\\mu^2}{d} = \\frac{T^2}{d^3}\n\n### Key insight\n\nPrecision depends only on the number of events d, not sample size or censoring pattern.\nThe coefficient of variation of \\hat\\mu is 1/\\sqrt{d}.\n\nA 95% CI is best formed on the log scale:\n\n\\exp\\\\!\\left(\\log\\hat\\mu \\pm \\frac{1.96}{\\sqrt{d}}\\right)",
  "title": "RMS Parametric Survival Models"
}