Question

why is this theorem for stats useful?

Answer 1

for mle that R0-R4 conditions are satisfied. Also that Fisher information satisfies, then any consistent sequence of solutions of mle equations satisfies this :
\[\sqrt{n}(\hat{\Theta} -\Theta)\rightarrow N(0,\frac{ 1 }{ I(\Theta) })\]

Answer 2

What are R0-R4? I assume that \(\Theta\) is a parameter and \(\hat{\Theta}\) is a consistent MLE taken from a sample \(X_1, X_2,..., X_n\).

Answer 3

i mean R0-R5. these are the conditions, for MLE, Fisher information and Cramer-Rao bound

Answer 4

regularity conditions

Answer 5

Could you list them here?

Answer 6

R0. If theta is not equal to theta(prime) then f(theta) and f(theta_prime) are different distributions

Answer 7

R1. The support of f(theta) i.e, supp(f(theta)):={x:f(theta(x)>0} is the same for all theta
R2. theta star is an interior point of omega

Answer 8

R3. F(theta x) is twice differentiable in theta for each x)

Answer 9

R4. integral of f(theta(x))dx in the continuos case is twice differentiable in theta

Answer 10

R5. thrice differentiable in theta for each x then there exists a constanc c>0 and a function M(X)>0

Answer 11

IIRC, the Fisher information is the variance of the score.
The theorem that you stated is a different form of the Central Limit Theorem. However, CLT deals with the sample itself, and the theorem you stated is used on an estimator of the sample.

Answer 12

how might it be important? for what?

Answer 13

For large enough n, you can use the asymptotic relationship to create confidence intervals and devise tests regarding the estimator.

Answer 14

Also, it may be the case that the pdf of the estimator is really complicated, such that deriving confidence intervals is hard. You can use the theorem you stated so that the pdf becomes asymptotic to normal.