MLE asymptotic properties in non-regular families I am working with asymptotic results about the MLEs and I know that if the family of distributions to whom the pdf of my sample belongs is exponential the regularity conditions for the asymptotic properties to hold are satisfied. 
Now, I am trying to generalize this result to non regular families (i.e. the parameter $\theta$ is in the support of $X$), but I am not sure which properties keep holding and which not. 
Someone can help me?
 A: "The parameter $\theta$ being in the support of $X$" creates regularity problems only for discrete random variables. For continuous random variables the regularity condition should be stated as "the support of $X$ does not depend on $\theta$" -the difference being that if $\theta$ is an interior point in the support, no harm done. In fact "does not depend on" is the more general description, encompassing both cases.
The classic example for a continuous random variable is the estimation of $\theta$ in a Uniform $[0,\theta]$. The support of $U$ depends on $\theta$. What we lose here is asymptotic normality, from the basic properties of the MLE. The problem (and the requirement) here is that $\theta$ is on the boundary.
By contrast, when we want to estimate the mean and the variance of, say, a normal distribution, both belong to the support of the variable, but being interior points to it, they do not create any problems.
On the other hand, if we have  a discrete random variable taking whole-number values in $\{1,2,\theta, 9, 10\},\; 2<\theta <9 $, possible problems may arise, even though $\theta$ does not affect the boundary of the support. 
This post may help you, https://stats.stackexchange.com/a/68866/28746 It contains basic papers for the properties of the MLE in non-standard conditions.
