Past exam of the statistics course of the University of Oxford / 2023 / 1 / b / i / Solution

Expanding the exponent, the log-likelihood function iswhere $C$ is independent of $\theta$. Its score function and second derivative areThe first is positive below $\bar{y}$ and negative above it, so the unique maximum-likelihood estimator isUsing $\mathbb{E}{\sum }_i{Y}_i=n\theta$, the one-observation Fisher information is $I(\theta )=-n^{-1}\mathbb{E}{\ell }^{\prime \prime }(\theta )={\theta }^{-3}$. The support does not depend on the interior parameter $\theta >0$, and the regular asymptotic normality of a maximum likelihood estimator givesThis is Inverse Gaussian mean estimation with known shape one. Equivalently, the Inverse Gaussian distribution has variance ${\theta }^3$, so the same result follows directly from the central limit theorem applied to its sample mean.