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Suppose we have multivariables ${\boldsymbol {\theta }}=\left[\theta _{1},\theta _{2},\dots ,\theta _{d}\right]^{T}\in {\mathbb {R}}^{d}$, and we want to estimate the function of parameters $\boldsymbol \phi(\boldsymbol {\theta })$ with unbiased estimator $\boldsymbol T(X)$, CRB states that we have a lower bound as: $${\displaystyle \operatorname {cov} _{\boldsymbol {\theta }}\left({\boldsymbol {T}}(X)\right)\geq \boldsymbol\phi (\theta )^TI\left({\boldsymbol {\theta }}\right)^{-1}\boldsymbol\phi (\theta )}$$ where $I(\boldsymbol \theta)$ is the fisher information matrix. I wonder what's the attainability of this bound, I find the bound in Wikipedia link but I can't find the attainability in the wekipedia link. Is this bound always attainable?

Thanks in advance!

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Under some conditions, the Cramer-Rao bound is attained if and only if we deal with an exponential family.

See e.g.

Lehmann, E. L. (1983). Theory of Point Estimation. Springer-Verlag, New York. p.123

or

Shao, Jun. Mathematical statistics. Springer Science & Business Media, 2003. p.171

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  • $\begingroup$ Thank you very much, but why I see in Wikipedia that the bound is always attainable by maximum likelihood estimator asymptotically? $\endgroup$
    – narip
    Oct 16, 2022 at 10:54
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    $\begingroup$ @narip The statement in Wikipedia is about asymptotics, $n\to\infty$, but the attainability theorem I cited above is for finite $n$. $\endgroup$
    – frank
    Oct 16, 2022 at 11:27

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