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Math Help - Identity Check

  1. #1
    Guy
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    Identity Check

    No one answers my questions, but I figure I might as well try.

    In solving a problem I've stumbled across something that I'd like to verify. Suppose X \sim N(0, \sigma^2 I_p) and Q is a symmetric p \times p matrix. Then, for any function f(\cdot) the following holds:

    E \left\{f(\|X\|^2) X^T Q X\right\} = E \left\{f(\|X\|^2) \|X\|^2 \right\} \frac{tr(Q)}{p}

    The idea behind this is to apply the spectral theorem to write \Gamma D \Gamma' and define Y = \Gamma' X. Then, Y \stackrel{d}{=} X and \|X\|^2 \equiv \|Y\|^2.

    From there, argue

    E \left\{f(\|X\|^2) X^T Q X\right\} <br />
= E \left\{f(\|Y\|^2) Y^T D Y\right\} <br />
= E \sum_{i = 1} ^ p \lambda_i Y_i ^ 2 f(\|Y\|^2)
    = \sum_{i = 1} ^ p \lambda_i E [Y_i ^ 2 f(\|Y\|^2)]<br />
= E [Y_i ^ 2 f(\|Y\|^2)] tr(Q)<br />
= \frac{tr(Q)}{p} \sum_{i = 1} ^ p  E [Y_i ^ 2 f(\|Y\|^2)]
    = E \left\{f(\|X\|^2) \|X\|^2 \right\} \frac{tr(Q)}{p}

    (The \lambda_i are the eigenvalues of Q if that isn't clear). I guess the main thing I need to get this off the ground is that f(\|Y\|^2) Y_i are identically distributed for all i. This seems true since I think (\|Y\|^2, Y_i) is identically distributed for all i. This justifies the steps where I do stuff with the expected values.
    Last edited by Guy; October 25th 2010 at 12:26 PM.
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  2. #2
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    i suspect not many people understand what you wrote. Or is that just me.
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by brennan View Post
    i suspect not many people understand what you wrote. Or is that just me.
    Is that Chinese?
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  4. #4
    Guy
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    Something is unclear? It's all completely standard matrix notation...tr(Q) is the trace of Q, X is a random vector of length p with the specified variance covariance matrix, Y is an appropriate orthogonal transformation of X, \Gamma D \Gamma^T is the spectral decomposition of Q...

    I mean, it's not intro level math-stat, but I don't think it's unreasonable for someone in statistics to read it and understand the notation. Maybe I'll stick with visiting my professor's office hours instead of hoping for faster answers here.
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  5. #5
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    Quote Originally Posted by Guy View Post
    Something is unclear? It's all completely standard matrix notation...tr(Q) is the trace of Q, X is a random vector of length p with the specified variance covariance matrix, Y is an appropriate orthogonal transformation of X...

    I mean, it's not intro level math-stat, but I don't think it's unreasonable for someone in statistics to read it and understand the notation. Maybe I'll stick with visiting my professor's office hours instead of hoping for faster answers here.
    Stick with it here - some one who studies this field will pop along soon. Its just not my field - but I look at many threads out of interest. I originally replied cos u where fed up about no one answering your questions. But if that got ur hopes up - sorry.
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