 Solution

Let $n=2m\ge 2$. Between the two middle order statistics, $k=m$ and the slope $A^{\prime }(\mu )=2m-n$ is zero. To the left $A$ decreases and to the right it increases. Consequently all joint maximum-likelihood estimates arewhere the constant minimum isThe strict positivity follows from distinct observations. Thus the location maximum-likelihood estimate is not unique, while the rate maximum-likelihood estimate is unique and the same for every location in the displayed interval. Choosing the arithmetic midpoint of the two middle values is the usual sample median convention and gives one member of this entire maximizing family.