There are so many possible answers and the subject is so large!
<p>When the parametric family of distributions is some exponential family, there is this recent and very elegant paper:

 - [Consistency under Sampling of Exponential Random Graph Models][1]

Although it is focused on studying exponential random graph models, which are commonly used in modeling social networks (and other network models), the framework is quite general.



  [1]: http://arxiv.org/abs/1111.3054