common support for MLE Why common support is important for MLE? What happen if a family of distributions 
fail to have common support? I would like to understand where and why in the proof of MLE properties (existense, consistency, effciency, asymptotic normality) we need the regularity condition of common support.
 A: Deriving the key properties of MLE (e.g. Cramer-Rao, asymptotic efficiency relative to M-estimators, etc) depends on being able to differentiate under the integral sign.
The validity of differentiating under expectation is usually established via dominated convergence theorem. When the distributions do not have common support, the dominated convergence argument may not apply and MLE results can break down.
Consider, for example, the property
$$
E_{\theta}[\frac{\partial}{\partial \theta} \log f(x;\theta)] = 0,
$$
which is needed for Cramer-Rao. This is obtained by differentiating $E_{\theta}[f(x;\theta)]$ under the expectation sign. (It also makes MLE a method of moment type estimator.)
Here is a counterexample where the distributions do not have common support.
Consider the family of uniform distributions parametrized by $\theta$: 
$f(x;\theta) = \frac{1}{\theta}1_{[0,\theta]}$.
$$
E_{\theta}[\frac{\partial}{\partial \theta} \log f(x;\theta)] = -\frac{1}{\theta} \neq 0,
$$
Cramer-Rao lower bound does not hold in this case. The dominated convergence type argument breaks down because $-\frac{1}{\theta}$ is not Lebesgue integrable.
