= 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 $Y=\mathbf1_{\{X>T\}}$ for a fixed $T>0$ instead gives a <Bernoulli distribution> with parameter $r=e^{-\lambda T}$. The chain rule for its <score function> gives
$$
I_Y(\lambda)=\frac{(r')^2}{r(1-r)}=\frac{T^2}{e^{\lambda T}-1},\qquad
\frac{I_Y}{I_X}=\frac{s^2}{e^s-1}<1,\quad s=\lambda T.
$$
To see the strict inequality, write $e^s-1=s+s^2/2+s^3/6+\cdots$. For $0<s\le1$ the first two terms already exceed $s^2$. For $s\ge1$, 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 $s\ge1$. 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.
Back to article page
