I am having a problem with a very basic concept in the minimum distance estimation approach in statistical inference.
I've read a paper which uses, for a parametric model family of discrete distributions, a common countable support as its very first step of the estimation procedure. However, if we just consider the family to contain all Binomial(n,.7) (just to fix p and vary n) or Discrete Uniform (1/n) (means assigning masses 1/n to each of the n points 1/n, 2/n,..., n/n and 0 elsewhere) distributions, where the supports themselves depend on the parameters, then it seems very difficult to extend the notion of the common family support to cover the such cases, where each family member has a different support, depending on its parameter.
Remember, in such cases, while finding out a suitable disparity between the empirical mass function (relative frequency) and the actual p.m.f., the determination of the actual pmf will create the central dificulty. I would love to see a general definition and approach covering all discrete distributions. Can anyone enlighten me?