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?

  • $\begingroup$ To be uncorrelated would mean that S and T would be independent $\endgroup$ – Jon Oct 25 '16 at 16:38
  • 2
    $\begingroup$ @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

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.


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