the MLE is which is an sample mean of Bernoulli's
So
and by Slutsky's Theorem
as well.
Hello all! Having a problem with general MLE concepts. See if you can help me with this.
Let X1,...Xn be a random sample of size n where Xi~BIN(1,p). How do I find the Asyptotic distribution of the MLE of p?
Thanks for any help!
That looks like the formula adapted to Bin(1,p) for n observations, or, for large n
(with MLE= and Fisher information for one observation, so for n observations)
So, does the former ( ) comes from Slutsky theorem? I just never seen this name before, and I was also given the above assymptotic formula without explaining where it comes from.
The Central Limit Theorem gives you
replacing the p's and q's with their sample versions also converges in distribution to
a standard normal since
and by either law of large numbers
all you need is convergence in probability http://en.wikipedia.org/wiki/Slutsky's_theorem
Ah! then, that assymptotic formula with Fisher information in it looks just like a version of the Central Limit Theorem. (edited: but of course it would, because it originates from CLT, I just didn't see the proof so I didn't see that it does. Sorry for off-top).