State cramr-rao inequality

1. How do you prove Cramer-Rao inequality?
2. Why do we use Cramer-Rao inequality?
3. What are the major assumption of CR inequality?
4. Can Cramer-Rao lower bound be negative?

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 do we use Cramer-Rao inequality?

The Cramér–Rao inequality is important because it states what the best attainable variance is for unbiased estimators. Estimators that actually attain this lower bound are called efficient. It can be shown that maximum likelihood estimators asymptotically reach this lower bound, hence are asymptotically efficient.

What are the major assumption 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 ∈ .

Can Cramer-Rao lower bound be negative?

If the data points are on average below the true population mean, then the score is negative.