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Math Help - Need help understanding consistency

  1. #1
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    Need help understanding consistency

    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 n 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 Y_1, Y_2,...,Y_n by a random sample of size n from a normal pdf having \mu=0. Show that S_n^2=\frac{1}{n}\sum_{i=1}^nY_i^2 is a consistent estimator for \sigma^2=Var(Y)

    I tried this:
    \lim_{n\to\infty}P(\frac{1}{n}\sum_{i=1}^nY_i^2-\sigma^2<\epsilon)=1

    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.
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  2. #2
    MHF Contributor matheagle's Avatar
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    This is just the Weak Law of Large Numbers

    {\sum_{i=1}^nW_i\over n}\buildrel P\over\to E(W)

    So {\sum_{i=1}^nY^2_i\over n}\buildrel P\over\to E(Y^2)=\sigma^2 since the MOO is zero
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  3. #3
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    sorry, what is MOO and what is \buildrel P\over\to? Is it a function P?
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  4. #4
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by jass10816 View Post
    sorry, what is MOO and what is \buildrel P\over\to? Is it a function P?
    MOO is the mean(ie) here
    and that symbol means convergence in probability.
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  5. #5
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    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 \frac{Var(S_n^2)}{\epsilon^2}<\delta for arbitrary \epsilon and \delta.

    Then, Var(S_n^2)=\frac{1}{n}Var(Y^2) and Var(Y^2)=E(Y^4)-E(Y^2)^2

    Since \mu=0, E(Y^4)=M_y^{(4)}(0)

    I found the fourth moment about the origin to be 3\sigma^4

    Then Var(Y^2)=3\sigma^4-\sigma^4=2\sigma^4

    So Var(S_n^2)=\frac{2\sigma^4}{n}

    So for large enough n, \frac{2\sigma^4}{n\epsilon^2}<\delta

    which shows that S_n^2 is consistent for \sigma^2
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