Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 218 / 6 / c / ii

A method-of-moments estimator usesUnder $|\theta |<1$, choose the invertible rootwith the continuous value zero when ${\hat{r}}_1=0$. Alternatively maximize the exact Gaussian likelihood using the positive-definite covariance matrix from part (i), producing ${\hat{\theta }}_{\mathrm{ML}}$. Given either estimate,is the moment estimate; likelihood estimation may instead profile ${\sigma }^2$. Under a fixed interior parameter $|\theta |<1$ and standard stationary ergodic finite-moment regularity, both estimators are consistent and asymptotically normal, with Gaussian maximum likelihood asymptotically efficient.