Does anyone know of a tried and true method for distinguishing local from global maxima when solving for the MLE?
This depends on what sort of problems you are looking at. What is the context? If you are taking an undergraduate or perhaps first year graduate course in statistical theory, there are reliable methods for getting MLE's. In full generality, however, it is not an easy problem. There are reasonably common models that don't yield to straight forward techniques when looking for the MLE. Optimization is a subject matter of considerable depth.
Specifically, I am interested in maximizing the MLE for various models, such as the Dagum or Singh-Maddala model, fitted to income distribution (grouped) data. There are only about 20 observations, so I don't think we can assume global concavity (can we?).
I tried a montecarlo heuristic along the lines of simulated annealing which worked pretty well, but does not reach an optimum where the gradient is 0.
I've heard of people solving this particular problem by quasi-Newton methods and by direct search, but my attempts with these methods have been completely unsuccessful so far.