Attainablility of Cramer Rao Bound with function of multi-parameters? 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!

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

