Exponential class of probability density functions?

Answer 1

The question is rather ... short. You are probably referring to the Regular Exponential Class (REC). A density function is said to be a member of REC if it can be decomposed as follows (LaTeX code, since I don't know how to write this stuff here):

f(x;s)=c(s)h(x)exp[\sum_{j=1}^{k} q_j(s) t_j(x)] for x \in A and zero otherwise. Here, s is a vector of k unknown parameters (s_1, ... , s_k). Moreover, the following conditions need to be satisfied: the parameter space needs to be compact (i.e. for all i, we have fore each parameter that a_i \leq s_i \leq b_i, for a_i and b_i constants, where also infinity is allowed), also, the set A over which the density is strictly positive should not depend on the parameters s, the functions q_j should be non trivial, functionally independent, and continuous functions of the parameter, and the last condition is that the derivatives t'_j(x) are linearly independent continuous functions of x over A (with a similar condition existing for discrete random variables).

Why we care about REC? Various reasons: e.g. we can then immediately derive a set of complete and sufficient statistics, or construct uniformly most powerful tests for one-sided hypothesis testing.