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Thread: Variance of a weighted function

  1. March 20th 2009, 06:27 AM #1
    markrvr
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    Variance of a weighted function

    Given
    \hat{f}(x)=\frac{1}{nh}\sum_{i=1}^{n}K\left(\frac{ x-X_i}{h}\right)
    and
    E\hat{f}(x)=\int_{-\infty}^{\infty}\frac{1}{h}K\left(\frac{x-y}{h}\right)f(y)dy
    show that
    \mbox{var}\hat{f}(x)=\frac{1}{n}\int_{-\infty}^{\infty}\frac{1}{h^2}K\left(\frac{x-y}{h}\right)^2f(y)dy-\left\{\frac{1}{h}\int_{-\infty}^{\infty}K\left(\frac{x-y}{h}\right)f(y)dy\right\}^2
    Last edited by Jhevon; March 20th 2009 at 11:49 AM. Reason: fixed LaTeX...whew! well, Moo helped out
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