Fisher information loss from thresholding an exponential observation

If $X$ has an exponential distribution with rate $\lambda >0$, a full observation has Fisher information $1/{\lambda }^2$. Recording only $Y=1_{\{X>T\}}$ for a fixed $T>0$ instead gives a Bernoulli distribution with parameter $r=e^{-\lambda T}$. The chain rule for its score function givesTo see the strict inequality, write $e^s-1=s+s^2/2+s^3/6+\cdots$. For $0<s\le 1$ the first two terms already exceed $s^2$. For $s\ge 1$, the function $e^s-1-s^2$ is positive at one, its derivative $e^s-2s$ is positive there, and that derivative is increasing for $s\ge 1$. Thus every positive threshold discards information. For independent observations, both information quantities multiply by the sample size. The Cramér-Rao bound and asymptotic normality of a maximum likelihood estimator explain the corresponding loss of attainable precision.