Cramr-rao lower bound proof

1. How do you derive Cramer-Rao lower bound?
2. How do you prove Cramer-Rao inequality?
3. Why we use Cramer-Rao lower bound?
4. Does MLE always achieve Cramer-Rao lower bound?

How do you derive Cramer-Rao lower bound?

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 .

How do you prove Cramer-Rao inequality?

Using the above proposition, we can now give a proof of the Cramér-Rao inequality for an arbitrary sample size n. E(VXi (θ)) = nE(VX(θ)) = 0. |E(V (θ) · ˆθ)| = |Cov(V (θ), ˆθ)| ≤ √ V ar(V (θ))V ar(ˆθ). V ar(VXi (θ)) = nI(θ).

Why we use Cramer-Rao lower bound?

The Cramer-Rao lower bound (CRLB) expresses limits on the estimate variances for a set of deterministic parameters. We examine the CRLB as a useful metric to evaluate the performance of our SBP algorithm and to quickly compare the best possible resolution when investigating new detector designs.

Does MLE always achieve Cramer-Rao lower bound?

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