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Assume independent survival times and positive rate $\lambda$. With $S={\sum }_i{t}_i$, the likelihood is $L(\lambda )={\lambda }^n{e}^{-\lambda S}$ on the observed nonnegative data. The score and negative expected second derivative are $n/\lambda -S$ and $n/{\lambda }^2$. For $S>0$, which holds almost surely under the model,The usual independent-sample likelihood asymptotics give $\sqrt{n}(\hat{\lambda }-\lambda )\Rightarrow N(0,{\lambda }^2)$. Thus an approximate Wald interval isFor a positive interval with the same first-order justification, apply the delta method to $\log \hat{\lambda }$, obtaining endpoints $\hat{\lambda }\exp (\pm z_{1-\alpha /2}/\sqrt{n})$.

A single individual survives until $T$ with probability $s=e^{-\lambda T}$. Thus $N_T\sim \mathrm{Bin}(n,s)$ and the new likelihood is $\left(\genfrac{}{}{0ex}{}{n}{n_T}\right)e^{-\lambda T{n}_T}(1-e^{-\lambda T}{)}^{n-n_T}$. For $0<n_T<n$, its maximum occurs at $s=n_T/n$, givingIf $n_T=0$, the supremum occurs as $\lambda \to \infty$ and there is no finite maximum. If $n_T=n$, it occurs at the boundary $\lambda =0$ of the extended parameter space, or is only a supremum when the rate is required to be strictly positive.

A binomial experiment has information $n/[s(1-s)]$ for $s$. Since $ds/d\lambda =-Ts$, parameter transformation givesAs quantified by Fisher information in an exponential survival snapshot, the ratio of this information to the complete-data information is $z^2/(e^z-1)$ with $z=\lambda T>0$, strictly less than one by the supplied inequality. Hence every fixed survival snapshot loses information. Under regular large-sample conditions its efficient rate estimator has larger asymptotic variance, and correspondingly wider uncertainty intervals, than the estimator based on all individual survival times. This comparison concerns information and asymptotic uncertainty, not a claim that every realized interval in one experiment is wider.