I have a small math problem
I have a sample of size N iid normal distributed variables. Then I get the MLE for mu and sigma^2(but the biased estimater for sigma^2, the one that is 1/N*(theRest) ). Now I have to prove consistecy of them both. Proving that for the unbiased estimator of mu is easy using Chebyshev.
My problem is proving the consistency of the biased MLE estimator of sigma^2.
Can anybody help me?
chebyshev is not the way to go.
you would need a fourth moment in that case
You can obtain STRONG consistency by the Strong Law of Large Numbers.
And almost sure implies convergence in prob, but we do have strong consistency here
AND I used a.s. which doesn't need cheby either.
But I can prove these with cheby, but you will need a fourth moment
And we don't need normality either here.