 Solution

The log density isIts kink occurs at the unknown location $\mu$, so its parameter dependence cannot be separated into a fixed finite-dimensional statistic of $y$ and the smooth canonical form required of an exponential dispersion family. Thus the two-parameter location-scale Laplace distribution is not such a family.

For independent observations, the log-likelihood isFor every fixed $\sigma >0$, maximizing this expression in $\beta$ is equivalent to minimizing $S(\beta )$. Hencethe least absolute deviations estimator. At this value,so, in the nondegenerate case $S(\hat{\beta })>0$,