Let
$$
\bar q=\frac1K\sum_{k=1}^Kq_k,
\qquad
\bar x=\frac1N\sum_{i=1}^Nx_i,
$$
and set
$$
A=\operatorname{Var}(\bar q)
=\sigma_\mu^2+\sigma_G^2+\frac{\sigma_I^2}{K},
\qquad
B=\operatorname{Var}(\bar x)
=\frac{\sigma_{\rm tot}^2}{N}.
$$
The within-galaxy contrasts $q_k-\bar q$ contain no $M_0$, so the parameter-dependent <log-likelihood> reduces to
$$
\ell(M_0,\theta)
=-\frac{(\bar q-M_0)^2}{2A}
-\frac{\{\bar x-(M_0-\theta)\}^2}{2B}+\text{constant}.
$$
The <score equations> give the <maximum-likelihood estimators>
$$
\widehat M_0=\bar q,
\qquad
\widehat\theta=\bar q-\bar x.
$$
The calibrator and Hubble-flow samples are independent, so both estimators are <unbiased estimator>[unbiased] and
$$
\operatorname{Var}(\widehat M_0)=A,
\qquad
\operatorname{Var}(\widehat\theta)=A+B.
$$
The <Fisher information matrix> for $(M_0,\theta)$ is
$$
\mathcal I=
\begin{pmatrix}
A^{-1}+B^{-1}&-B^{-1}\\
-B^{-1}&B^{-1}
\end{pmatrix},
\qquad
\mathcal I^{-1}=
\begin{pmatrix}
A&A\\
A&A+B
\end{pmatrix}.
$$
Hence $\operatorname{Var}(\widehat\theta)=(\mathcal I^{-1})_{22}$: $\widehat\theta$ attains the multiparameter <Cramer-Rao bound>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 219 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-219/1/c/solution