I'm trying to work on some sufficiency problems and am coming to the following conclusion - I'd love to have someone give me the thumbs up if it looks good or let me know if it's wrong. Thanks!

In evaluating sufficiency (not minimal sufficiency, just sufficiency), it seems that I can use three tools:

1) Solve directly and see if I get a data reduction (such as using indicator functions that reduce...)

2) Factorization theorem, in which we get h(x) * g(theta, T(x)) where T(x) is sufficient.

3) Theorem for minimal sufficient statistics looking at f(x|theta) / f(y|theta). If the ratio equals a constant for some x=y (or function of x=y), then those are sufficient statistics. (Am I correct that in this case, we would not have to prove that it's only sufficient in that particular case when x=y, since we are not looking for minimal sufficiency?)

Does that look right? Thanks!!!