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This question links to a document by Jon Wellner that defines the sufficient statistic for the multivariate normal (p. 7, Example 2.7). The result follows from the factorization theorem and is proven in a response to the post.

However, Example 5 (pp. 1316, 1320-1321) in this article by Ghosh et al (2010), a review of ancillarity and conditional inference, implies that a sufficient statistic for the correlation under bivariate normality is not known. Specifically, it shows there are multiple, non-unique ancillary complements for the correlation under bivariate normality. It gives numerous examples from the literature of proposed ancillary complements. And, it implies that selecting or constructing the best possible ancillary statistic to condition on, remains an open (or perhaps unsolvable) problem.

My confusion is that I'd think the sufficient statistic for the bivariate normal distribution would imply a sufficient statistic for the correlation, given that the latter is a parameter of the former. Is this not so? If not, why not? If so, why condition on an ancillary complement? Or am I just misinterpreting the examples--say, is the statistic given by Wellner only asymptotically sufficient, or only theoretical?

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There's no problem with finding a sufficient statistic for the correlation, and the any sufficient statistic for the bivariate Normal is one such.

In fact, the Ghosh et al paper says in their Example 5 that $\{ U_1+U_2, W\}$ is minimal sufficient, where $U_1$ is the sum of squares of $X$ and $U_2$ the sum of squares of $Y$, and $W$ the sum of products. The issue is that this minimal sufficient statistic is two-dimensional and the correlation is a single parameter, so it would be nice to identify some one-dimensional function of $\{ U_1+U_2, W\}$ (or some two-dimensional function of $\{U_1, U_2, W\}$) as an ancillary complement. That's what they can't do.

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  • $\begingroup$ Thanks! I clearly overlooked the phrase "minimal sufficient" in the Ghosh paper. Can you explain why it would be "nice" to do so? Is it purely aesthetic, or is there some practical case where it would be more useful to have an ancillary in the same number of dimensions as the parameter? $\endgroup$
    – virtuolie
    Commented May 20 at 17:58
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    $\begingroup$ It's not having an ancillary of the same size as the parameter, it's having the non-ancillary part the same size as the parameter. That is, we'd ideally like a one-dimensional minimal sufficient statistic for a one-dimensional $\rho$. Failing that, if we have a $k$-dimensional sufficient statistic we'd like $k-1$ dimensions of it to be ancillary, so there's one 'informative' dimension in some sense for the single parameter. This isn't of any great practical importance; it's more a question of how powerful 'ancillary' and 'sufficient' are as concepts in simple parametric models. $\endgroup$ Commented May 21 at 0:16
  • $\begingroup$ Actually, a second motivation is hinted on p 1321: "Either of these [alternative] ancillary statistics [for the correlation] can be used for higher order approximations to the distribution of the maximum likelihood estimator [of the correlation]...." The authors say earlier (p. 1310) that "...when the MLE...itself is not sufficient, there is loss of Fisher information," but conditioning on a true ancillary can make the MLE sufficient. So, although this isn't explicit, I think there's a loss of information by using U1+U2, which is only approximately ancillary. $\endgroup$
    – virtuolie
    Commented May 21 at 17:57

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