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In these lecture notes:

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However I am unable to work this out myself. When I multiply things out I get $\frac{1}{nS_{xx}}(\sum_{j=1}^n x_j^2 -2n\bar{x}x_i+nx_i^2)$. For things to be true, the terms inside the parenthesis can be rearranged to be $S_{xx}+n(x_i-\bar{x})^2$. I tried rearranging the terms so $\sum_{j=1}^n x_j^2 -n\bar{x}x_i+nx_i^2-n\bar{x}x_i$, but I can't seem to get to the answer. There must be some form of $S_{xx}$ that I am not aware of that is buried in their somewhere. Any help would be appreciated.

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  • $\begingroup$ Have you used the relation $S_{xx}= x_i^2-n\bar{x}^{2}$? $\endgroup$ – felasfa Oct 9 '16 at 22:31
  • $\begingroup$ If you use that, you get the resulting equation. Let me know otherwise $\endgroup$ – felasfa Oct 9 '16 at 22:35
  • $\begingroup$ I assume you mean $S_{xx}=\sum x_i^2 - n \bar{x}^2$. There is no $\bar{x}^2$ anywhere, so I am not sure where I would use that formula. Thanks! $\endgroup$ – Carl Ganz Oct 9 '16 at 22:43
  • $\begingroup$ yes, you are right. I forgot the summation. $\endgroup$ – felasfa Oct 9 '16 at 23:16
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After multiplying things out, one gets \begin{align} h_{ii}=\frac{1}{nS_{xx}}\left(\sum_{j=1}^{n} x_{j}^{2}-2n\bar{x}x_{i}+nx_{i}^{2}\right)&=\frac{1}{nS_{xx}}\left(\sum_{j=1}^{n} x_{j}^{2}-n\bar{x}^{2}+[n\bar{x}^{2}-2n\bar{x}x_{i}+nx_{i}^{2}]\right)\\ &=\frac{1}{nS_{xx}}\left(\sum_{j=1}^{n} x_{j}^{2}-n\bar{x}^{2}\right)+\frac{1}{nS_{xx}}[n\bar{x}^{2}-2n\bar{x}x_{i}+nx_{i}^{2}]\\ & = \frac{1}{nS_{xx}}\cdot S_{xx} + \frac{1}{nS_{xx}}n[x_{i}-\bar{x}]^{2}=\frac{1}{n}+\frac{1}{S_{xx}}[x_i-x]^{2} \end{align}

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\begin{align} h_{ii}&=\frac{1}{nS_{xx}}(\sum_{j=1}^n x_j^2 -2n\bar{x}x_i+nx_i^2+n\bar{x}^2_n-n\bar{x}^2_n)\\ &=\frac{1}{nS_{xx}}\left((\sum_{j=1}^n x_j^2 -n\bar{x}^2_n)+n(-2\bar{x}x_i+x_i^2+\bar{x}^2_n)\right)\\ &=\frac{1}{n}+\frac{(x_i-\bar{x}_n)^2}{S_{xx}} \end{align}

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