# Does Basu's Theorem require minimal sufficiency?

Casella & Berger state Basu's Theorem (Th 6.2.24) as follows:

If $T(X)$ is a complete and minimally sufficient statistic, then $T(X)$ is independent of every ancillary statistic.

However, in lecture, I saw a proof of the theorem that used only sufficiency, not minimal sufficiency. The proof was basically an application of the law of total probability.

Wikipedia states Basu's Theorem using sufficiency and bounded completeness (a weaker requirement than completeness), which agrees with my lecturer.

What gives with the Casella-Berger version?

• Regarding Wikipedia's proof, remember Bahadur's Theorem: If $T$ is a boundedly complete sufficient statistic and finite-dimensional, then it is minimal sufficient. – Zen Feb 20 '16 at 16:23
• I see. So the version my lecturer presented breaks only in the case that is not boundedly complete. Thank you! – half-pass Feb 20 '16 at 19:31

To realise that sufficiency is not enough, consider that, when $T(X)$ is a sufficient statistic, $(T(X),S(X))$ is also a sufficient statistic. Including the case when $S(X)$ is an ancillary statistic. Meaning that $(T(X),S(X))$ and $S(X)$ cannot be independent.