= Solution
Assume <independent> survival times and positive rate $\lambda$. With $S=\sum_it_i$, 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,
$$
\boxed{\widehat\lambda=\frac nS,\qquad I_{\mathrm{full}}(\lambda)=\frac n{\lambda^2}.}
$$
The usual <independent>-sample <likelihood> asymptotics give $\sqrt n(\widehat\lambda-\lambda)\Rightarrow N(0,\lambda^2)$. Thus an approximate Wald <interval> is
$$
\boxed{\widehat\lambda\pm z_{1-\alpha/2}\frac{\widehat\lambda}{\sqrt n}.}
$$
For a positive <interval> with the same first-order justification, apply the <delta method> to $\log\widehat\lambda$, obtaining endpoints $\widehat\lambda\exp(\pm z_{1-\alpha/2}/\sqrt n)$.
A single individual survives until $T$ with probability $s=e^{-\lambda T}$. Thus $N_T\sim\operatorname{Bin}(n,s)$ and the new <likelihood> is $\binom n{n_T}e^{-\lambda Tn_T}(1-e^{-\lambda T})^{n-n_T}$. For $0<n_T<n$, its <maximum> occurs at $s=n_T/n$, giving
$$
\boxed{\widetilde\lambda=-\frac1T\log\frac{n_T}n.}
$$
If $n_T=0$, 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 gives
$$
\boxed{I_T(\lambda)=\frac{nT^2s}{1-s}=\frac{nT^2}{e^{\lambda T}-1}.}
$$
As 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.
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