Partial sufficiency with connection to the identification problem

Let M^Θ_X = (R_X ,X,P^Θ = {P^θ: θ ε Θ}) be a parametrized statistical model and g: Θ → G be a non-injective function characterizing a parameter of interest. The basic idea of partial sufficiency is to find a (minimal) statistic sufficient for making inference on g(θ). Following Fraser (1956), Barndorff-Nielsen (1978) has defined a concept of S-sufficiency. Our contribution is first to establish the connection between S-sufficiency and the identification concept. Second, we establish some properties of S-sufficiency, in particular we compare the properties of sufficiency for the complete parameter with those of S-sufficiency.