How does knowing the sign of the population correlation affect the sufficiency of its statistic?

Question

As noted here, the sufficient statistic for the correlation under bivariate normality is Pearson's $r$, the maximum likelihood estimate of $\rho$. I suppose, however, this does not guarantee that $r$ will be sufficient when one knows the sign of $\rho$, i.e., $\rho \geq 0$ or $\rho \leq 0$. In that case, but still under bivariate normality, is $r$ in fact still sufficient? Say, after truncating at 0? If not, does a sufficient statistic exist?

Answer 1

A fast and general answer. Let us first remember the definition of sufficiency, here I take it from Bickel & Doksum: Let $\mathcal{P}$ be a parametric family of distributions, parameterized by $\theta \in \Theta$. Then $T(X)$ is called sufficient for $\mathcal{P}$ (or for $\theta$) if the conditional distribution of $X$ given $T(X)=t$ does not involve $\theta$.

In your case the parameter space for $\rho$ is $[-1, 1]$. If any statistic is sufficient for $\rho$ with that parameter space, it will continue to be sufficient for a reduced parameter space, like $[0,1]$. If the conditional distribution of the data given some statistic does not involve $\rho$, when considering the full space, restrictions on the parameter space cannot induce dependence on $\rho$!

This is very general, and have nothing in particular to do with normal distributions or correlations, but is a general mathematical fact.