= Fisher information in an exponential survival snapshot
{c}
{title2=$I_T(\lambda)=nT^2/(e^{\lambda T}-1)$}
For $n$ <independent> <exponential distribution>[exponential] lifetimes with rate $\lambda$, recording all lifetimes gives <Fisher information> $n/\lambda^2$. Recording only the count alive at a fixed time $T>0$ gives a <binomial distribution> with success probability $s=e^{-\lambda T}$. Its information in $s$ is $n/[s(1-s)]$, and the change of parameter $ds/d\lambda=-Ts$ gives the displayed information.
The retained fraction is $R(z)=z^2/(e^z-1)$, $z=\lambda T$. It is strictly below one, so every single snapshot loses information. The inequality follows, for instance, from $h(z)=e^z-1-z^2$: its <derivative> is $e^z-2z>0$ for positive $z$, because that <derivative> has its <local minimum>[minimum] at $z=\log2$ with value $2-2\log2>0$. Also $R(z)$ tends to zero as $z\downarrow0$ and as $z\to\infty$.
Differentiating $R$ shows that the best snapshot time solves $2(1-e^{-z})=z$. The difference of the two sides is strictly concave, starts at zero with positive <derivative>, and eventually becomes negative, so it has exactly one positive root, approximately $1.59362$. The largest retained fraction is about $0.64761$. Thus even an optimally timed snapshot keeps less than two thirds of the full lifetime information. This is a local design result: choosing that time requires a nominal rate or a preceding estimate.
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