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!

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- Mar 16th 2011, 09:46 PMcangerAsymptotic distributions of MLEs
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! - Mar 16th 2011, 11:43 PMmatheagle
the MLE is which is an sample mean of Bernoulli's

So

and by Slutsky's Theorem

as well. - Mar 17th 2011, 07:12 AMVolga
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. - Mar 17th 2011, 06:27 PMmatheagle
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 - Mar 17th 2011, 10:04 PMVolga
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).