Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 1

LetSubtracting the measured distance modulus from the measured apparent magnitude givesFor the Hubble flow, defineThe Hubble law, with the Hubble constant parametrized by $\theta =5{\log }_{10}h$, gives $r_i\sim N(M_0-\theta ,H_i)$. After integrating over each intrinsic absolute magnitude and each unobserved true distance modulus, independence therefore gives the likelihood function

Put $w_k=C_k^{-1}$, $u_i=H_i^{-1}$, $A={\sum }_k{w}_k$, $B={\sum }_i{u}_i$, and form the weighted meansThe two equations obtained from the score function areConsequently the maximum-likelihood estimators areThe Hessian matrix of the log likelihood isIts first leading principal minor is negative and its determinant is $AB>0$, so it is a negative-definite matrix. Thus the stationary point is the unique global maximum.

Under homoskedasticity, writeThen ${\hat{M}}_0=\bar{q}$ and $\hat{\theta }=\bar{q}-\bar{r}$. Both are unbiased estimators, and their covariance matrix isIndeed the Fisher information isand its inverse is exactly the displayed covariance matrix. The estimators therefore attain the multivariate Cramer-Rao bound and are efficient estimators.

By the invariance property of maximum likelihood estimation,The sampling distribution is $\hat{\theta }\sim N(\theta ,{\sigma }_{\hat{\theta }}^2)$, where$\hat{h}$ has a log-normal distribution withWriting $v={\sigma }_{\hat{\theta }}^2/{\alpha }^2$, its exact fractional bias and variance areHence, to leading order,