# If the Cramer Rao's Lower Bound tends to zero, is the estimator efficient?

I apply a nonlinear transformation to a linear estimator $$\hat{\alpha}=f(\hat{\theta})$$. Then I find Cramer Rao's Lower Bound, and it asymptothically goes to zero. Does it means that my new estimator $$\hat{\alpha}$$ is efficient? Why?

I guess I should find $$E(\hat{\alpha})$$ and then see that $$E(\hat{\alpha}) \rightarrow E(\hat{\theta})$$, but I think it is not the point of the excercise, because it is a quite complicated nonlinearity.

According to Cramer-Rao lower bound we may write: $$\text{Var}(\hat{\alpha})\geq \frac{\left(1+\frac{db}{d\alpha}\right)^2}{nI(\theta)}$$ where $$b$$ is the bias of estimator. It can be easily seen that the lower bound may go to zero due to the factor $$n$$ in the denumerator.
In order to show that an estimator is efficient, you should show that it is unbiased and also attains the lower bound in the Cramér–Rao inequality, for all $$\theta$$. This means that you should show that:
$$\text{Var}(\hat{\alpha})= \frac{1}{nI(\theta)}$$