I am having trouble with the proof of Basu's theorem... specifically, I'm not sure about the $\theta$s in the expectations below:
Let $T$ be a complete sufficient statistic. Let $V$ be an ancillary statistic. Let $A$ be an event in the sample space.
Basu's theorem states that $V$ and $T$ are independent. We need to show:
$\mathbb{P}( V \in A | T )$ $=$ $\mathbb{P}(V \in A)$
So, $\mathbb{P}(V \in A)$ $=$ $\mathbb{E}[I(V \in A)]$
$=$ $\mathbb{E}_{\theta}[(I(V \in A)]$ (Question: Why is $\theta$ here if we're talking about an ancillary statistic?)
=$\mathbb{E}_{\theta}\mathbb{E}_{\theta}[I(V \in A)|T]$
$=$ $\mathbb{E}_{\theta}\mathbb{E}[I(V \in A)|T]$ (Question: I understand that the $\theta$ disappears from the second expectation here since T is a sufficient statistic?)
From this we conclude $\mathbb{E}_{\theta}[g(t)$ $-$ $\mathbb{P}(V \in A)]$ $=$ $0$ for all $\theta$ in the sample space. (Queston: Why is $g(t)$ subtracted from $\mathbb{P}(V \in A)$ here? Why are we concluding from the above that the expectation is 0?
Thus $\mathbb{E}_{\theta}[I(V \in A)|T]$ $=$ $\mathbb{P}(V \in A)|T)$ $=$ $\mathbb{P}(V \in A)$