When does a maximum likelihood estimate fail to exist? I have been told that a maximum likelihood estimate (MLE) does not always actually exist.  Why is this the case?  It is clear that the MLE may not be unique, but there should always be a maximum, no? 

The MLE exists if the parameter space is compact and the Likelihood function is continuous on the parameter space.  
It is unique if the parameter space is convex and the likelihood function is concave.  
Edit 
I should add that the above conditions are sufficient conditions.
