# Unbiased estimator based on minimal sufficient statistic has smaller variance than one based on sufficient statistic

Suppose that $T_1$ is sufficient and $T_2$ is minimal sufficient, U is an unbiased estimator of $\theta$, and define $U_1=\mathbb{E}(U|T_1)$ and $U_2=\mathbb{E}(U|T_2)$

a)Show that $U_2=\mathbb{E}(U_1|T_2)$

b) Now use the conditional variance formula to show that $\text{Var}U_2 \leq \text{Var}U_1$

So I am having some trouble even getting started and partially looking for some hints where to begin. I think I have the differences between minimal and sufficient understood, but still a bit shaky.

I've been trying to go though what I know about conditional expectation and trying to apply and manipulate $U_1$ and $U_2$ and hopefully getting both parts to = U and see if that does anything. Is there a property of conditional expectation that I am just missing or am I missing something about how expectation works with minimal/sufficient statistics?

• If $T_2$ is minimal sufficient and $T_1$ sufficient, what does this tell you of the connections between those two statistics? – Xi'an Mar 19 '15 at 20:31
• Hint: can you compare $\mathbb{E}(U|T_1)$, $\mathbb{E}(U|T_1)$, and $\mathbb{E}(U|T_1,T_2)$ when $T_2=f(T_1)$? – Xi'an Mar 20 '15 at 7:48
• Is this related to using the ratio of $f(x|\theta)$ and $f(y|\theta)$ (in the general sense)? – James Snyder Mar 21 '15 at 23:44