 Solution

The log-likelihood function for the Laplace distribution isThus maximum likelihood estimation is equivalent to minimizing the sum of absolute values. Write $n=2m+1$ and sort the data as $Y_{(1)}\le \cdots \le Y_{(n)}$. Between observations, if $r$ values lie to the left of $\kappa$, the derivative of ${\sum }_i|Y_i-\kappa |$ is $r-(n-r)=2r-n$. It is negative before the middle observation and positive after it. The minimum occurs at the sample median:This minimizer remains unique for odd $n$ even when some observations coincide, since fewer than half the observations can lie strictly on either side of the middle value.