Past exam of the options I course of the University of Oxford / 2013 / O / 1 / a / Solution

Put $S={\sum }_{j=1}^n|X_j|$. The log-likelihood for the Laplace distribution isFor $S>0$ the derivative is positive for $\theta <S/n$ and negative for $\theta >S/n$, so the unique maximum-likelihood estimator isThe exceptional data set $S=0$ has probability zero; there the likelihood has its supremum at the excluded boundary $\theta =0$. For the Fisher information, $|X_j|$ has exponential density ${\theta }^{-1}{e}^{-y/\theta }$ for $y>0$, hence mean $\theta$. Since ${\ell }^{\prime \prime }(\theta )=n/{\theta }^2-2S/{\theta }^3$,This also follows from the score variance, because $\mathrm{Var}(S)=n{\theta }^2$.