May I ask one statistics related question please? I have one query on MLE itself. It's property says that, for large sample size it is normally distributed [i.e. asymptotically normal]. On the other hand it is Efficient as well. My doubt is, how this two asymptotic properties exist simultaneously? If it is consistent then asymptotically it should collapse to "truth" i.e. for large sample size, variance of MLE should be zero. However asymptotic normality says, MLE have some distribution and hence variance.
 
Can anyone please clarify me? Your help will be highly appreciated.
 


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