Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|improve this question
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 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 at 19:31
up vote 9 down vote accepted

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.