1. Parameter Estimation Proof

Hi there

Im trying to do a proof from a past paper at uni, but I'm not sure how to go about it. The proof is this below:

Prove, for any smooth probability density function f(x|\theta ), that:

\hat{\theta} D\to N({\theta}_{0}; \frac{1}{nI({\theta}_{0})})

(The D is meant to be on the arrow but I dont know how to get it there, think it means converges in distribution)

as n \to \infty

where {\theta}_{0} is the true value of the parameter \theta .
Hint: You may use the fact that the expectation of the score function is zero.

Its very similar to a proof from our textbook and i sort of understand the logic, but I'm not sure how to set it out for a normal distribution. I can prove that under smoothness the MLE of theta hat is consistent, which seems quite similar.. except that its as P\to f(x|\theta ) but I cant quite figure out how the two are related :/

I think I have to take MLE of the normal and use the weak law of large numbers to get the expected value of it and then do some manipulation, but i cant figure out where the Fisher information comes into play.
Im pretty hopeless at stats :/ Help would be much appreciated. Sorry if my post seems confusing, I'm quite new here, will be happy to clarify anything.

2. Put $$...$$ around your latex for it to be interpreted by MHF.

Hi there

Im trying to do a proof from a past paper at uni, but I'm not sure how to go about it. The proof is this below:

Prove, for any smooth probability density function f(x|\theta ), that:

$\hat{\theta} D\to N({\theta}_{0}; \frac{1}{nI({\theta}_{0})})$

(The D is meant to be on the arrow but I dont know how to get it there, think it means converges in distribution)

as n \to \infty

where ${\theta}_{0}$ is the true value of the parameter \theta .
Hint: You may use the fact that the expectation of the score function is zero.

Its very similar to a proof from our textbook and i sort of understand the logic, but I'm not sure how to set it out for a normal distribution. I can prove that under smoothness the MLE of theta hat is consistent, which seems quite similar.. except that its as P\to f(x|\theta ) but I cant quite figure out how the two are related :/

I think I have to take MLE of the normal and use the weak law of large numbers to get the expected value of it and then do some manipulation, but i cant figure out where the Fisher information comes into play.
Im pretty hopeless at stats :/ Help would be much appreciated. Sorry if my post seems confusing, I'm quite new here, will be happy to clarify anything.
Ugh, such a sloppily worded question. The fact that the MLE of theta hat is consistent, which you can prove to yourself, makes it obvious that what is asserted in the question statement is FALSE: $\hat \theta$ converges in distribution to a POINT MASS at $\theta_0$.

What they mean, of course, is that $\sqrt n (\hat \theta - \theta_0) \to N(0, I^{-1}(\theta_0)).$ The idea is to first show that the score function evaluated at $\theta_0$ converges by the CLT to $N(0, I(\theta))$ and then do a Taylor expansion on the score function about $\theta_0$. This of course completely ignores how you deal with the remainder term of the Taylor expansion, which is the hard part of the problem, but hey, who cares about trifling things like that?

I'm like 90% sure you need a lot more than smoothness to get this result to go through, if you want to do it rigorously. You need a lot of stuff to deal with the remainder term of the Taylor expansion. Even in Casella and Burger they just hand-wave their way through dealing with the remainder term (though they at least mention the need for a lot of regularity conditions).

4. Thanks for all the help guys, you'll really pushed me to thinking in the right direction. Was quite a tricky 12 mark proof, i.e. 12% of the paper.