 Solution

For each fixed $\lambda >0$, maximizing the likelihood function over $\mu$ is equivalent to minimizing $A(\mu )={\sum }_i|x_i-\mu |$. Order the distinct observations as $x_{(1)}<\cdots <x_{(n)}$ and write $n=2m+1$. Between successive order statistics, with exactly $k$ observations to the left,Thus $A$ decreases up to $x_{(m+1)}$ and increases after it. Its unique minimum is at the sample median:At a fixed $\mu$ with $A(\mu )>0$, the log-likelihood function has derivative $n/\lambda -A(\mu )$ and second derivative $-n/{\lambda }^2<0$, so its unique maximizing rate is $n/A(\mu )$. The maximized likelihood function is $(n/[2eA(\mu )]{)}^n$, which increases when $A(\mu )$ decreases. Hence the joint maximum-likelihood estimates areThis is Laplace location and scale maximum likelihood, expressed with the reciprocal scale.

A finite rate maximum-likelihood estimate requires a positive total absolute deviation. For distinct observations and odd $n\ge 3$, that condition holds. For $n=1$, however, $\mu =x_1$ gives $A(\mu )=0$ and $L=\lambda /2\to \infty$ as $\lambda \to \infty$. There is no finite joint maximum-likelihood estimate in the stated parameter space; the formula with zero denominator is not an admissible estimate.