Can the numerical variance be lower than Cramer-Rao Lower Bound?

1. What is the Cramer Rao lower bound for the variance of unbiased estimator of the parameter?
2. What is the relationship between the Fisher information and the Cramer Rao lower bound?
3. What are the major assumptions of CR inequality?
4. How do you prove Cramer Rao inequality?

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.

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.

What are the major assumptions of CR inequality?

One of the basic assumptions for the validity of the Cramér–Rao inequality is that the integral on the left hand side of the equation given above can be differentiated with respect to the parameter θ under the integral sign. As a consequence, it is as follows. ˆθ(x) f (x,θ)dx = θ, θ ∈ .

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).