I read the textbook in Cramer-Rao lower bound (CRLB). Here is a theroem

For some $\tau=\tau(\theta)$, there exists an unbiased estimator $\hat{\tau}$ of $\tau$ such that $Var(\hat{\tau})$ attains the CRLB if and only if the distribution belongs to an exponential family ($X\sim f(x; \theta)=\exp(a(x)b(\theta)+c(x)+d(\theta))$).

I know how to prove that if the distribution belongs to an exponential family, then there exists an unbiased estimator $\hat{\tau}$ of $\tau$ such that $Var(\hat{\tau})$ attains the CRLB.

But how to prove the another direction?

For some $\tau=\tau(\theta)$, there exists an unbiased estimator $\hat{\tau}$ of $\tau$ such that $Var(\hat{\tau})$ attains the CRLB, then the distribution belongs to an exponential family

I read the textbook in Cramer-Rao lower bound (CRLB). Here is a theroem

For some $\tau=\tau(\theta)$, there exists an unbiased estimator $\hat{\tau}$ of $\tau$ such that $Var(\hat{\tau})$ attains the CRLB if and only if the distribution belongs to an exponential family ($X\sim f(x; \theta)=\exp(a(x)b(\theta)+c(x)+d(\theta))$).

I know how to prove that "if the distribution belongs to an exponential family, then there exists an unbiased estimator $\hat{\tau}$ of $\tau$ such that $Var(\hat{\tau})$ attains the CRLB".

But how to prove the another direction?

For some $\tau=\tau(\theta)$, there exists an unbiased estimator $\hat{\tau}$ of $\tau$ such that $Var(\hat{\tau})$ attains the CRLB, then the distribution belongs to an exponential family.