double if not exponential

In one of my last quizzes for the year, as the course is about to finish, I asked whether mean or median was the MLE for a double exponential sample of odd size, without checking for the derivation of the result, as I was under the impression it was a straightforward result. Despite being outside exponential families. As my students found it impossible to solve within the allocated 5 minutes, I had a look, could not find an immediate argument (!), and used instead this nice American Statistician note by Robert Norton based on the derivative being the number of observations smaller than θ minus the number of observations larger than θ.  This leads to the result as well as the useful counter-example of a range of MLE solutions when the number of observations is even.

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