= 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 proves
$$
\boxed{\frac{\sqrt n(\widehat\theta-\theta)}{\theta}\ \xrightarrow{d}\ N(0,1).}
$$
Thus the large-sample approximation is $\widehat\theta\approx N(\theta,\theta^2/n)$, consistent with inverse <Fisher information>. Also $\widehat\theta\to\theta$ in <probability> by the <weak law of large numbers>, so $\theta$ in the standard error can be replaced by $\widehat\theta$ through <Slutsky's theorem>.
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