Let $X_1,X_2,...,X_n$ (assume $n\geq 2$) be a random sample from an $N(\mu,\sigma^2)$ population where $-\infty<\mu <\infty$ and $\sigma^2>0$ are unknown. Which of the following statements is/are true ?
(A) The MLE of $\mu$ attains Cramer-Rao lower bound. TRUE (because, unbiased)
(B) The UMVUE of $\mu$ attains Cramer-Rao lower bound. TRUE (because, unbiased)
(C) The MLE of $\sigma^2$ is an unbiased estimator of $\sigma^2$. FALSE (denominator is 1/n)
(D) The relative efficiency of the MLE of $\sigma^2$ w.r.t the UMVUE of $\sigma^2$ is strictly less than 1. TRUE (because UMVUE has efficiency of 1 and MLE has efficiency lower than 1)
I need some validation to ensure my reasons and answers are correct here. Please advise.
A. This is true because a normal pdf satisfies the conditions of the Cramer-Rao theorem and $\bar{x}$ (the MLE) is complete and sufficient for $\mu$ and is unbiased for $\mu$.
B. This is true for the same reasons as in A, since the UMVUE is identical to the MLE in this case.
C. This is false. It can be shown that the expected value of the MLE is $\frac{n-1}{n}\sigma^2$.
D. This is false. While the MLE is biased for $\sigma^2$, it has a lower variance which results in a better relative efficiency than the UMVUE. The relative efficiency can be easily calculated using that $Var(S_n) = \frac{2\sigma^4}{n-1}$, $Bias(\hat{\sigma^2},\sigma^2)=-\frac{1}{n}\sigma^2$ and that $\hat{\sigma^2} = \frac{n-1}{n}S_n$.