 Solution

The absolute observations are independent exponential random variables with mean $\theta$ and variance ${\theta }^2$. Applying the central limit theorem directly to their average provesThus the large-sample approximation is $\hat{\theta }\approx N(\theta ,{\theta }^2/n)$, consistent with inverse Fisher information. Also $\hat{\theta }\to \theta$ in probability by the weak law of large numbers, so $\theta$ in the standard error can be replaced by $\hat{\theta }$ through Slutsky's theorem.