Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 219 / 1

An apparent magnitude is the sum of absolute magnitude and distance modulus. ThusThe independent unbiased distance-modulus estimate is $N({\mu }_{\mathrm{LMC}},{\sigma }_{\mathrm{LMC}}^2)$, so closure of independent normal distributions under linear combinations givesHence $\mathbb{E}{\hat{S}}_1=M_0^{\ast }$ and $\mathrm{Var}({\hat{S}}_1)=V_1$.

The common distance modulus of calibrator galaxy $g$ cancels between its supernova and mean Cepheid apparent magnitudes. Their independent intrinsic scatters therefore giveThus $\mathbb{E}{\hat{S}}_2^g=\Delta M$ and $\mathrm{Var}({\hat{S}}_2^g)=V_2$.

The Hubble law gives $d_i=c{z}_i/(100h)$ Mpc, so its distance modulus isConsequently

All three sets of statistics are independent. With $s_1={\hat{S}}_1$, $s_{2g}={\hat{S}}_2^g$, and $s_{3i}={\hat{S}}_3^i$, their likelihood function, up to a parameter-independent factor, is

Differentiating the log-likelihood gives the unique stationary pointIts Hessian is diagonal with entries $-1/V_1$, $-G/V_2$, and $-N_{\mathrm{SN}}/{\sigma }_{\mathrm{SN}}^2$, so it is the unique maximum. The estimators are unbiased and have varianceswhich equal their Cramér-Rao lower bounds because the normal location statistics are efficient.

Since $\theta =M_0^{\ast }+\Delta M-\mathcal{M}$, invariance of the maximum-likelihood estimator givesIt is unbiased and normal with variance

By invariance of maximum likelihood,Because $\hat{\theta }-\theta \sim N(0,{\sigma }_{\theta }^2)$, the ratio $\hat{h}/h$ is log-normal. Its exact variance isAs $N_{\mathrm{SN}},G,N_{\mathrm{LMC}}\to \infty$, all sampling terms vanish but $V_1$ retains ${\sigma }_{\mathrm{LMC}}^2$. The dominant remaining uncertainty is therefore the parallax calibration of the LMC distance modulus.