Local maxima anomalies of likelihood methods Likelihood methods have many desired properties. Sadly, local maxima in finite samples is not one of them. The fact a local maximum exists near the true parameter value is of no comfort if one converges to different local maxima. 
I am looking for works reviewing this problem, demonstrating it, and offering solutions. 
 A: As Cagdas Ozgenc points out in his answer, there is a simple sufficient condition to render this question moot: that the likelihood be concave in the parameter space (almost surely with respect to the sampling distribution of the data).  That does cover many interesting cases (ie, the exponential family), but basically leaves everything else out.
I don't have an answer here, but think that there are several ways this question could be refined or restated:


*

*What properties does the MLE have in finite samples, and under what models?  Although everyone likes the MLE, it's usage (AFAIK) is predicated on asymptotic guarantees.  I can't think of any finite sample guarantees for it.

*What properties, if any, does a local maxima have?


Bibliography
"Evaluation of the Maximum-Likelihood Estimator where the Likelihood Equation has Multiple Roots" VD Barnette 1966.
The Cauchy distribution with location parameter offers a canonical example of a likelihood with multiple roots (even asymptotically).
"Testing for a Global Maximum of the Likelihood" Christophe Biernacki, 2005.
A test for consistency for a root of the likelihood equation, based on comparing the observed maximized likelihood to its expected value under the putative argmax
"Eliminating Multiple Root Problems in Estimation" Small, Wang, Yang 2000.
If you are going to read one paper, this is probably it.  Discusses all of the above, also in the context of generalized estimating equations, plus suggests smoothing or penalizing the likelihood to help resolve multiple roots.
A: No estimator will converge in probability to the parameter value in finite samples due to the random nature of data in general. Maximum likelihood estimators are consistent and efficient among unbiased estimators of parameter values. In fact, MLEs are root-n consistent by the CLT, meaning that they converge to the parameter value as the sample size increases considerable faster than any other general unbiased estimator (in most cases) including method of moments. Simply put, in correct specifications of probability models for data generating mechanisms, MLE is one of the best tools around. If you're looking for a general probability theory text dealing with asymptotic estimation, I'd refer you to Lehmann & Casella's Theory of Point Estimation.
A: This is not an issue of finite samples. It is related to the complexity of the combined error function and the link function. As long as it is quadratic there won't be multiple maxima. If not you will most likely get more than one. For example if you are hypothesizing that the data has a normal distribution then you have:
$N(f(y|x;\theta),\sigma)$
where f is the link function. Now if you consider the pdf:
$f(y) = \frac{1}{(2\pi)^{1/2}\sigma} e^{-\frac{(y-f(x;\theta))^2}{2\sigma^2}}$
The combined function, link function plugged into the distribution, in this scenario is quadratic (don't worry about the exponential as it disappears when you work with its logarithm) if $f(y|x;\theta)$ is a linear function. This happens in linear regression with Gaussian errors for example.
In any case if your distribution doesn't have a quadratic form or when combined with the link function ends up non-quadratic you will have the problem of multiple maxima.
There is no good solution to the problem as finding global maximum is NP-hard. 
http://web.maths.unsw.edu.au/~rsw/lgopt.pdf
pages 20-21
There are algorithms that attempt to cover the entire parameter space, but they are very slow as one expects. So it is not an issue of MLE either. If you plug the Likelihood function to a more sophisticated optimizer like simulated annealing, it will search for the global solution.
http://en.wikipedia.org/wiki/Simulated_annealing
