Writing $r_s=1/\alpha_s$, the log likelihood is
$$
\ell(r_0)=\text{constant}-3N\log r_0-\frac1{r_0}\sum_sr_s.
$$
Hence
$$
\boxed{\widehat r_0=\frac1{3N}\sum_{s=1}^N\frac1{\alpha_s}}.
$$
The score variance and negative expected Hessian give <Fisher information>
$$
\mathcal I_N(r_0)=\frac{3N}{r_0^2}.
$$
Since $r_s$ is Gamma with mean $3r_0$ and variance $3r_0^2$,
$$
\mathbb E\widehat r_0=r_0,\qquad
\operatorname{Var}(\widehat r_0)=\frac{r_0^2}{3N}.
$$
The estimator is unbiased and attains the <Cramér-Rao lower bound>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-219/1/c/solution