3
$\begingroup$

By Basu's theorem, we know that any ancillary statistic is independent of a statistic that is both sufficient and complete. I was wondering if the assumption of sufficiency and completeness can be relaxed.

If $T$ is a complete statistic for a family of distributions $\mathcal P = \{ P_{\theta} , \theta \in \Theta \}$, then for any ancillary statistic $S$, can we show that $S $ and $T$ are uncorrelated?

$\endgroup$
  • $\begingroup$ To be uncorrelated would mean that S and T would be independent $\endgroup$ – Jon Oct 25 '16 at 16:38
  • 3
    $\begingroup$ @Jon No, lack of correlation does not imply independence. $\endgroup$ – dsaxton Oct 25 '16 at 16:40
  • $\begingroup$ @dsaxton, that is correct. I take that comment back. $\endgroup$ – Jon Oct 25 '16 at 16:43
  • $\begingroup$ What do you mean? Complete but not sufficient? Do such a beast exist? $\endgroup$ – kjetil b halvorsen Feb 10 at 18:09
  • $\begingroup$ On the other hand if two statistics are independent then they are uncorrelated. So what are you looking for in the question? $\endgroup$ – Michael Chernick Feb 10 at 18:36
0
$\begingroup$

Statistical Inference by Casella Berger states a theorem

Theorem 6.2.28 : If a minimal sufficient statistic exists, then any complete statistic is also a minimal sufficient statistic.

Also under mild conditions, a minimal sufficient statistic does always exist.Which makes a complete statistic minimal sufficient via the above mentioned theorem. In particular, these conditions always hold if the random variables (associated with Pθ ) are all discrete or are all continuous.

Therefore, we can show that a complete statistic S is uncorrelated to any ancillary statistic T.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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