# Solved – Basu’s Theorem Proof

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 $$mathbb{E}_{theta}[I(V in A)|T]$$ $$=$$ $$mathbb{P}(V in A)|T)$$ $$=$$ $$mathbb{P}(V in A)$$

Contents

\$E_{theta}E_{theta}[I_{V in A}|T]\$

is taken "with a fixed T = t", giving a function \$g(t)=E_{theta}[I_{V in A}|T=t]\$, as Xi'an said. The second \$E_{theta}\$ then takes the expectation of g(t) (so you vary t now).

From this we conclude \$E_{theta}[g(t) − P(V in A)] = 0\$ for all θ in the sample space. (Queston: Why is \$g(t)\$ subtracted from \$P(Vin A)\$ here? Why are we concluding from the above that the expectation is 0?

This is using the definition of a complete statistic:

If you have a function h(T) that

• 1) does not depend on the parameter \$theta\$ directly, but only on T (as it is written, "h(T)", and
• 2) for which \$E_{theta}(h(T)) = 0\$ for whatever \$theta\$ you pick,

then \$h(t)\$ is itself zero almost everywhere (or: \$P_{theta}(h(T) = 0) = 1\$), again for any value of \$theta\$.

In the proof, T is complete, and the function h of T is \$h(T) = g(T) − P(V in A) = g(T) – c\$. We need that V is ancillary because else, h would not be purely a function of T, but some \$h(theta, T)\$.

"From this we conclude …": your first five lines of formulas say that \$E_{theta}(I_{V in A}) = E_{theta}[g(T)]\$,so \$ E_{theta}[h(T)]\$ is their difference and zero, so the second point is fulfilled, too, so the conclusion follows.

I know it's over two years late. I just figured this out for myself (or so I think). My problem was overlooking the first requirement. Hope it makes sense.

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