Usefulness of Point Estimators: MVU vs. MLE In a past class, two types of point estimators were introduced: minimum variance unbiased estimators (MVUs) and maximum likelihood estimators (MLEs). Supposedly, the MVU is optimal, unless an unbiased estimator doesn't exist.
In that class, we also extended the Neyman-Pearson Lemma to composite hypotheses, replacing unknown parameter values with their maximum likelihood values (why not MVU?), within the the constraints of the hypotheses. This means, unless the MLE is outside the parameter space, the MLE is used as the point estimate.
Why do we care about maximizing probabilities when calculating the Neyman-Pearson likelihood ratio? Since it's not used in hypothesis testing, what applications would use the MVU instead of the MLE? How important is it really for the estimator to be unbiased, since it doesn't seem to be an issue with hypothesis testing?
 A: You seem to have a bit of a misconception. MVU's and MLE's are not mutually exclusive.
A Maximum Likelihood Estimator is the statistic $\hat\theta_{mle}$ which maximizes the Likelihood function. That is
$$\hat\theta_{mle} = \arg\max_\theta L(\theta \vert X)$$
A Minimum Variance Unbiased Estimator is the statistic $\hat\theta_{mvu}$ with the properties.
$$E(\hat\theta_{mvu}) = \theta$$
$$ \text{For all $T$ such that $E(T) = \theta$, } \ \ Var(\hat\theta_{mvu}) \leq Var(T)$$

There is no reason that an estimator $\hat\theta$ cannot be both the MVU and the MLE. For instance, if $X_1, X_2, \cdots X_n$ are iid Exponential random variables with mean $\theta$, then the mle is $\hat\theta_{mle} = \bar X$.  
Clearly this estimator is unbiased (i.e. $E(\bar X) = \theta$), and the variance is
$$Var(\hat\theta_{mle}) = Var(\bar X) = \frac{\theta^2}{n}$$
It turns out however, that the Cramer Rao Lower Bound for the exponential distribution here is precisely $\frac{\theta^2}{n}$. To see this, we first find Fishers Information $I(\theta) = -E\left(\frac{\partial^2\ell(\theta|X)}{\partial\theta^2}\right)$.
$$\ell(\theta|X) = -n\log\theta + \frac{\sum_{i=1}^n x_i}{\theta}$$
$$\frac{\partial^2\ell(\theta|X)}{\partial\theta^2}= \frac{2\sum_{i=1}^nX_i}{\theta^3} - \frac{n}{\theta^2}$$
$$I(\theta) = E\left(\frac{2\sum_{i=1}^nX_i}{\theta^3} - \frac{n}{\theta^2}\right) = \frac{2n\theta}{\theta^3} - \frac{n}{\theta^2} = \frac{n}{\theta^2}$$
The Cramer-Rao Lower Bound states that for any estimator $\hat\theta$ which is unbiased for $\theta$, it is the case that $Var(\hat\theta) \geq \frac{1}{I(\theta)}$. Since $\bar{X}$ acheives the CRLB, this guarantees that $\bar{X}$ has minimum variance of all unbiased estimators for $\theta$. Thus $\hat\theta_{mle} = \bar{X} = \hat\theta_{mvu}$. It is actually fairly common for the MLE to be the MVU!

It is difficult to define exactly what is meant by optimal for estimation, as it depends on your goal. I think almost everybody would agree that MVU estimators are optimal among the class of unbiased estimators. But often Mean Square Error is used as a criteria for "goodness" of an estimator, and usually leads to "optimal" estimators which are slightly biased. MLE's have lots of provably good properties, which is largely why Neyman Pearson theory relies on them, but I think that belongs in a separate question.
