As the title, can the moment estimator sometimes be NOT sufficient? Consider the following example where Y ~ Uniform(1-b, 5+b):
From method of moment estimator:
E[Y] = 1/2((1-b) + (5+b)) = 3 => Not useful
E[Y^2] = 1/12 (5+b - 1+b)^2 + 3^2= 1/12(2b+b)^2 + 9
Therefore, we set the second moment estimator m'2 to equal E[Y^2] and solve for b:
b = 1/2(sqrt(12(m'2 - 9)) - 4)
However, when we determine the sufficient stats by factoring, we will obtain that the first order Y(1) and last order Y(n) are minimally sufficient, meaning that m'2 is not sufficient. This seems weird to me because I though Moment estimators should always be sufficient. Is this true? Or did I do something wrong?