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Thread: Variance estimation

  1. #16
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    Re: Variance estimation




    Quote Originally Posted by Masteras View Post
    for n=2 the have the same variance the V and S^2_{n-1}.
    Z=X-Y~N(0,2*sigma^2). from theory, we know it.
    From which theory? Are you using mgf for transformation?

    Transformations for W=Z^2 to end up with a gamma. The U=W/a (where a=2sigma^2) to see what kind of a gamma is this (chi-square actually). I told you to take it step by step to understand. keep in mind that the variance of your V (for n=2) is 2sigma^4, the same.
    I am sure you have something interesting but I can't follow you. It's easier to see when you actually write some mathematical arguments rather than "keep in mind that the variance of your V (for n=2) is 2sigma^4, the same"
    Last edited by _joey; 11-01-2010 at 05:07 AM.

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    Re: Variance estimation

    Here is the theory.

    If X_iare independent, standard normal random variables then the sum of their squares will follow chi-square distribution with k degrees of freedom.

    A random variable can be standardised. Since we have N~(0,\sigma^2) then we also have

    \frac{X}{\sigma},\frac{Y}{\sigma} which are standard normal random variables and the sum of their square will follow chi-square distribution. It follows from this that \frac{X^2+Y^2}{\sigma^2} ~\chi^{2}_{2}; and
    Var(\frac{X^2+Y^2}{\sigma^2})=4

    A bit of algebra and you get Var(V)=\sigma^4

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    Re: Variance estimation

    let's see.
    Z=0.5*X^2+0.5*Y^2=X^2, since they are iid variables. X~N(0,\sigma^2).
    The distribution of this Z is a Ga(1/2,2\sigma^2). The variance of a Ga(a,b) is ab^2.

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    Re: Variance estimation

    BGM in this post also thinks variance for S^{2}_{n-1} is higher.
    http://talkstats.com/showpost.php?p=40161&postcount=9

    Anyway, thanks for your replies but I can't see why variances for two estimators are the same.

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    Re: Variance estimation

    Since you opu it this way, take BGM's 1st hint in that post and work the distribution of V, construct V from the first hint. The distribution is a Ga(1,\sigma^2). The variance of this is ...

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    Re: Variance estimation

    Quote Originally Posted by _joey View Post
    I think I've solved it. I should write I used your idea to solve the problem. Var(V)=\sigma^4 and Var(s^2)=2\sigma^4 where n=2

    I've got a quick question: is there an easy way to show \frac {(X - Y)^2} {2\sigma^2} \sim \chi^2(1)?

    Err I mean for n, not just n=2.
    The generalization is

    \frac {S^2_{n-1}} {\sigma^2} \sim \chi^2(n-1)

    Is that what you want?

  7. #22
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    Re: Variance estimation


    The proof of general form. I will search books.google.com

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