This is just the Weak Law of Large Numbers
So since the MOO is zero
I am having a hard time understanding the way my textbook explains how to check consistency. I understand the concept of how it refers to the shape of the pdf of an estimator as a function of and the two examples are clear, but as soon as I try an exercise I am lost.
The first one I am looking at is
Let by a random sample of size from a normal pdf having . Show that is a consistent estimator for
I tried this:
I am not sure how to write this probability though. I tried to write it as an integral, but then I don't know how to solve the integral I got. I am really just guessing at this point, so if I could have any advice that would be great.
So I don't see how your first line implies the second line of your solution. Here's what I did if you don't mind checking my work:
I tried to show that for arbitrary and .
Then, and
Since ,
I found the fourth moment about the origin to be
Then
So
So for large enough n,
which shows that is consistent for