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Math Help - Kernel Distribution - show it is a density

  1. #1
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    Kernel Distribution - show it is a density

    Given the kernel density estimator \hat{f} (s)=\sum^{N}_{n=1} \frac{1}{N\cdot \delta} K\left( \frac{s-x_n}{\delta}\right), show that it is a density for every N, and every \delta>0 and every relization of the sample ie. show the funciton is nonnegative and integrates to one.

    I'm not sure how to formalise thise, I know that if K(\cdot) is a density, then it integates to one by definition. Does the following work;
    \int^{\infty}_{-\infty}\hat{f}(s)ds=\int^{\infty}_{-\infty}\sum^{N}_{n=1} \frac{1}{N\cdot \delta}  K\left(  \frac{s-x_n}{\delta}\right)ds=\frac{1}{N\cdot \delta}  \int^{\infty}_{-\infty}\sum^{N}_{n=1} K\left(  \frac{s-x_n}{\delta}\right)ds

    So I know that
    \int^{\infty}_{-\infty}\sum^{N}_{n=1} K\left(   \frac{s-x_n}{\delta}\right)ds needs to equal  N\delta but I'm not sure what to do with the integral..
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Robb View Post
    Given the kernel density estimator \hat{f} (s)=\sum^{N}_{n=1} \frac{1}{N\cdot \delta} K\left( \frac{s-x_n}{\delta}\right), show that it is a density for every N, and every \delta>0 and every relization of the sample ie. show the funciton is nonnegative and integrates to one.

    I'm not sure how to formalise thise, I know that if K(\cdot) is a density, then it integates to one by definition. Does the following work;
    \int^{\infty}_{-\infty}\hat{f}(s)ds=\int^{\infty}_{-\infty}\sum^{N}_{n=1} \frac{1}{N\cdot \delta} K\left( \frac{s-x_n}{\delta}\right)ds=\frac{1}{N\cdot \delta} \int^{\infty}_{-\infty}\sum^{N}_{n=1} K\left( \frac{s-x_n}{\delta}\right)ds

    So I know that
    \int^{\infty}_{-\infty}\sum^{N}_{n=1} K\left( \frac{s-x_n}{\delta}\right)ds needs to equal  N\delta but I'm not sure what to do with the integral..
    Observe that if u=(s-x_n)/\delta :

     <br />
\int_{-\infty}^{\infty}K\left(\frac{s-x_n}{\delta}\right)\;ds=\int_{-\infty}^{\infty}\delta K(u)\;du=\delta<br />

    So:

    \int_{-\infty}^{\infty}\hat{f} (s)\; ds=<br />
\int_{-\infty}^{\infty}\left[\sum^{N}_{n=1} \frac{1}{N\cdot \delta} K\left( \frac{s-x_n}{\delta}\right)\right]\; ds <br />
=\sum^{N}_{n=1}\left[ \int_{-\infty}^{\infty}\frac{1}{N\cdot \delta} K\left( \frac{s-x_n}{\delta}\right)\; ds\right]<br />
<br />
=\sum^{N}_{n=1}\frac{1}{N}=1

    CB
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