Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 29K (Principles of Statistics)

Continuous differentiability gives the first-order expansionThe assumed convergence in distribution implies $\sqrt{n}({\hat{\theta }}_n-{\theta }_0)=O_p(1)$, so after multiplication by $\sqrt{n}$ the remainder is $o_p(1)$. The Slutsky theorem gives the multivariate delta method

For observations $X_1,\dots ,X_n$, the log-likelihood is, up to an additive constant,Its unique maximum isNow $\mathbb{E}[X_i^2]={\sigma }^2$ and, using $\mathbb{E}[Z^4]=3$, $\mathrm{Var}(X_i^2)=2{\sigma }^4$. The central limit theorem followed by the delta method for $g(x)=\sqrt{x}$ gives

The one-observation score function isIts variance, or equivalently minus the expected second derivative of the log-likelihood, is the Fisher information

Since $\rho ={\sigma }^2$ and the reparametrization is one-to-one, invariance of the maximum-likelihood estimator gives