Does an Maximum Likeihood estimator always unique? It is true that an MLE is always unique for exponential family of probability distributions?

Question

The maximum likelihood estimator is the maximizer of the probability distribution for a sample drawn from an unknown distribution.

Is it unique?

If we consider uniform distribution then it is not unique. What will happen if we consider an exponential family of distributions? What is the MLE of the exponential family?

Answer 1

There is an article at The American Statistician addressing the issue you are concerned about:

Sudhakar Dharmadhikari & Kumar Joag-Dev (1985) Examples of Nonunique Maximum Likelihood Estimators, The American Statistician, 39:3, 199-200

And yes, the log-likelihood for the exponential family is a concave function, so if there is a solution for the first order conditions, this solution is a global max.