Is Complete Statistic Uncorrelated with Ancillary Statistic

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?

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