Fisher information and cramer-rao lower bound

1. What is the relationship between the Fisher information and the Cramer-Rao lower bound?
2. Does MLE always achieve Cramer-Rao lower bound?
3. What is the Cramer-Rao lower bound for the variance of unbiased estimator of the parameter?
4. Why is the Cramer-Rao lower bound important?

What is the relationship between the Fisher information and the Cramer-Rao lower bound?

In estimation theory and statistics, the Cramér–Rao bound (CRB) expresses a lower bound on the variance of unbiased estimators of a deterministic (fixed, though unknown) parameter, the variance of any such estimator is at least as high as the inverse of the Fisher information.

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.

Why is the Cramer-Rao lower bound important?

One of the most important applications of the Cramer-Rao lower bound is that it provides the asymptotic optimality property of maximum likelihood estimators. The Cramer-Rao theorem involves the score function and its properties which will be derived first.