There are a lot of problems that are solved by Maximum likelihood method. How to determine if the problem has a closed-form solution or not?
closed as too broad by kjetil b halvorsen, StasK, Xi'an, gung♦, John Oct 2 '15 at 16:25
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I doubt there is a rule here but in general this can be determined from the First Order conditions. If you end up having to solve non-linear equations, which cannot be simplified, then a numerical procedure is the way to proceed.