An estimator whose variance attains Cramer-Rao lower bound is consistent

Question

If the variance of an estimator attains the Cramer-Rao lower bound then the estimator is

$(A)$ most efficient

$(B)$ sufficient

$(C)$ consistent

$(D)$ admissible

I can show that the the variance is most efficient. But I'm unable to test whether it is consistent or NOT.

Please help me.

Answer 1

Given an ubiased estimator $\hat\theta$ of the parameter $\theta\in\Theta$, this estimator is efficient if its var-cov matrix equals the Cramér-Rao lower bound. In other words, the Cramér-Rao inequality provides a lower bound for the var-cov matrix of unbiased estimators. As you know, unbiasedness is different from consistency.

To show the consistency of $\hat\theta$, one must show that $plim\hat\theta$ = $\theta_0$. If the set $\Theta$ is compact; the objective function $\mathbb{Q}_0(\theta)$ is continuous and has a unique maximum in $\theta_0$, and; $\widehat{\mathbb{Q}_n}(\theta)$ converges (uniformily) in probability to $\mathbb{Q}_0(\theta)$; then $plim\hat\theta=\hat\theta_0$.

Lets consider the case of a linear model $Y_i=X_i\theta$ + $\epsilon_i$. The OLS estimator or MLE is $\hat\theta=\theta_0+(X'X)^{-1}X'\epsilon$. If one assumes that plim $n^{-1}(X'\epsilon)=0$ and that $\lim_{n\to \infty}n^{-1}X'X=Q$ where $Q$ is not singular, then $\hat\theta\rightarrow_p\theta_0$.