Given a family of distributions parametrized by $\theta$ for which Cramer-Rao bounds on variance for a (biased or unbiased) estimator of $\theta$ exist, these bounds may be unattainable. In the proof for the 1D case, the looseness is introduced by applying the Cauchy-Schwarz inequality on the covariance between the score and the statistic. Is there a “onto” map $\phi\mapsto\theta$ for which CRLBs for $\phi$ are tight?
2026-03-30 23:21:25.1774912885
Can reparameterization make Cramer-Rao bounds tight?
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Almost by tautology, such a reparameterization exists iff the set of distributions for the family in question is identical to that of a family possessing a minimal sufficient statistic.
Exponential families are some of the families that have minimal sufficient statistics, (though not all of them).
A deck that contains this entire discussion is found here: https://spinlab.wpi.edu/courses/ece531_2011/9crlb.pdf