Asymptotic cramer-rao lower bound

1. Why is Cramer-Rao lower bound?
2. How is Cramer-Rao lower bound calculated?
3. Does MLE always achieve Cramer-Rao lower bound?
4. What is the Cramer-Rao lower bound for the variance of unbiased estimator of the parameter?

Why is Cramer-Rao lower bound?

The Cramer-Rao Lower Bound (CRLB) gives a lower estimate for the variance of an unbiased estimator. Estimators that are close to the CLRB are more unbiased (i.e. more preferable to use) than estimators further away.

How is Cramer-Rao lower bound calculated?

Alternatively, we can compute the Cramer-Rao lower bound as follows: ∂2 ∂p2 log f(x;p) = ∂ ∂p ( ∂ ∂p log f(x;p)) = ∂ ∂p (x p − m − x 1 − p ) = −x p2 − (m − x) (1 − p)2 .

Does MLE always achieve Cramer-Rao lower bound?

The mle does not always satisfy the condition so the CRLB might not be attainable..

What is the Cramer-Rao lower bound for the variance of unbiased estimator of the parameter?

The function 1/I(θ) is often referred to as the Cramér-Rao bound (CRB) on the variance of an unbiased estimator of θ. I(θ) = −Ep(x;θ) ∂2 ∂θ2 logp(X;θ) . and, by Corollary 1, X is a minimum variance unbiased (MVU) estimator of λ.