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  "body": "\n## Definition\n\nLet $(\\tilde t_1, \\delta_1), \\dots, (\\tilde t_n, \\delta_n)$ be right-[[censored|MY8ZZP84NP]] observations of a lifetime, with $\\delta_i = 1$ for an event and $\\delta_i = 0$ for a censored time.\nLet $t_{(1)} \u003c \\dots \u003c t_{(m)}$ be the distinct event times, $d_j$ the number of events at $t_{(j)}$, and $n_j = \\#\\set{i : \\tilde t_i \\ge t_{(j)}}$ the number at risk just before $t_{(j)}$.\nThe *Nelson–Aalen estimator* of the cumulative [[hazard function|Y959NKQ3XW]] $H$ is\n$$\\hat H(t) = \\sum_{j \\,:\\, t_{(j)} \\le t} \\frac{d_j}{n_j}, \\qquad t \\ge 0,$$\nwith the empty sum equal to $0$.\n\n## Remarks\n\nIn counting process notation, with $N(t)$ the number of observed events up to $t$ and $Y(t)$ the number at risk at $t$, the [[estimator|0SSS6ERQ0S]] is $\\hat H(t) = \\int_0^t \\frac{dN(s)}{Y(s)}$, with $0/0 = 0$.\nA common variance estimator is $\\sum_{j : t_{(j)} \\le t} d_j / n_j^2$.\nThe survival estimate $e^{-\\hat H(t)}$ is at least the Kaplan–Meier estimate, since $e^{-x} \\ge 1 - x$ for all $x$.\n\nExample: for the observations $2, 3^+, 5, 7$, with $3^+$ censored, $(n_j, d_j)$ is $(4, 1)$ at $2$, $(2, 1)$ at $5$ and $(1, 1)$ at $7$.\nSo $\\hat H = 0$ on $[0, 2)$, $1/4$ on $[2, 5)$, $3/4$ on $[5, 7)$ and $7/4$ from $7$ on.\n\nNon-example: the cumulative sum $\\sum_{t_{(j)} \\le t} d_j / n$, which divides by the initial sample size $n$, is not the Nelson–Aalen estimator.\nIt ignores that subjects leave the risk set, and it underestimates the cumulative hazard.\n",
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