I already understood that because you explain it in the text; the further doubt I have is: concerning only the “only if” part, given a D-dimensional sufficient statistic exists by assumption, if ∑i∉Egi(xi) is also a D-dimensional sufficient statistic or not.
I think not, because it should not be able to capture what goes on with the E variables, that’s hidden in the completely arbitrary ∏i∈EP[Xi|Θ] term.
This annoys me because I can’t see the form of the sufficient statistic like in the i.i.d. case.
I already understood that because you explain it in the text; the further doubt I have is: concerning only the “only if” part, given a D-dimensional sufficient statistic exists by assumption, if ∑i∉Egi(xi) is also a D-dimensional sufficient statistic or not.
I think not, because it should not be able to capture what goes on with the E variables, that’s hidden in the completely arbitrary ∏i∈EP[Xi|Θ] term.
This annoys me because I can’t see the form of the sufficient statistic like in the i.i.d. case.